| L(s) = 1 | − 2·2-s + 2·4-s + 8·5-s − 4·7-s − 2·8-s − 3·9-s − 16·10-s + 2·11-s + 8·14-s + 3·16-s + 4·17-s + 6·18-s + 16·20-s − 4·22-s + 26·25-s − 8·28-s + 8·29-s + 6·31-s − 8·32-s − 8·34-s − 32·35-s − 6·36-s − 16·40-s + 10·43-s + 4·44-s − 24·45-s − 14·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 3.57·5-s − 1.51·7-s − 0.707·8-s − 9-s − 5.05·10-s + 0.603·11-s + 2.13·14-s + 3/4·16-s + 0.970·17-s + 1.41·18-s + 3.57·20-s − 0.852·22-s + 26/5·25-s − 1.51·28-s + 1.48·29-s + 1.07·31-s − 1.41·32-s − 1.37·34-s − 5.40·35-s − 36-s − 2.52·40-s + 1.52·43-s + 0.603·44-s − 3.57·45-s − 2.04·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57289761 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57289761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6165195474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6165195474\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T + p T^{2} + p T^{3} + T^{4} + p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} - 20 T^{3} + 199 T^{4} - 20 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 14 T^{2} + 15 p T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 52 T^{3} + 322 T^{4} - 52 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 302 T^{4} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 204 T^{3} + 2303 T^{4} - 204 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 238 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 - 2578 T^{4} + p^{4} T^{8} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 - 61 T^{2} + p^{2} T^{4} ) \) |
| 47 | $D_4\times C_2$ | \( 1 + 14 T + 98 T^{2} + 476 T^{3} + 2143 T^{4} + 476 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 6291 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 14934 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 30 T + 364 T^{2} + 30 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 10 T + 50 T^{2} + 540 T^{3} + 5207 T^{4} + 540 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 3912 T^{3} + 48782 T^{4} + 3912 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.30933697545596573452190316370, −10.10061249837486690201644991736, −9.966799080181860087922124247450, −9.814650105017873269838357365275, −9.270109868568830920065365797262, −9.107373645246503834989891941575, −8.997725565556758223409424030795, −8.891769209738381454171773928662, −8.205513294881170822683702853789, −7.995041530977554559415892509637, −7.60733016100094693303832992509, −6.99389594309050790637232112913, −6.79892353540196167752284573515, −6.38157430625912180181853927702, −5.97885165605733304586375633935, −5.92873622170064140322417833512, −5.85397264750666204541989304119, −5.48014038382242070525636287404, −4.62713367547717085035316920673, −4.49001075244372937160418627747, −3.19299315586788972126544937353, −3.00401276796830364195469335788, −2.87098581846513060130011001291, −1.69430857905798018070785768556, −1.69391131513832317371120585058,
1.69391131513832317371120585058, 1.69430857905798018070785768556, 2.87098581846513060130011001291, 3.00401276796830364195469335788, 3.19299315586788972126544937353, 4.49001075244372937160418627747, 4.62713367547717085035316920673, 5.48014038382242070525636287404, 5.85397264750666204541989304119, 5.92873622170064140322417833512, 5.97885165605733304586375633935, 6.38157430625912180181853927702, 6.79892353540196167752284573515, 6.99389594309050790637232112913, 7.60733016100094693303832992509, 7.995041530977554559415892509637, 8.205513294881170822683702853789, 8.891769209738381454171773928662, 8.997725565556758223409424030795, 9.107373645246503834989891941575, 9.270109868568830920065365797262, 9.814650105017873269838357365275, 9.966799080181860087922124247450, 10.10061249837486690201644991736, 10.30933697545596573452190316370
