| L(s) = 1 | + 4·4-s − 8·9-s − 8·11-s + 12·16-s − 20·25-s − 32·36-s + 24·43-s − 32·44-s + 32·64-s + 32·81-s + 64·99-s − 80·100-s − 24·107-s − 32·113-s + 32·121-s + 127-s + 131-s + 137-s + 139-s − 96·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 96·172-s + ⋯ |
| L(s) = 1 | + 2·4-s − 8/3·9-s − 2.41·11-s + 3·16-s − 4·25-s − 5.33·36-s + 3.65·43-s − 4.82·44-s + 4·64-s + 32/9·81-s + 6.43·99-s − 8·100-s − 2.32·107-s − 3.01·113-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 8·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 7.31·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8783391725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8783391725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_4\times C_2$ | \( 1 + 8 T^{2} + 32 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_4\times C_2$ | \( 1 - 48 T^{2} + 1152 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 24 T^{2} + 288 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_4\times C_2$ | \( 1 - 96 T^{2} + 4608 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_4\times C_2$ | \( 1 + 120 T^{2} + 7200 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_4\times C_2$ | \( 1 - 48 T^{2} + 1152 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_4\times C_2$ | \( 1 + 72 T^{2} + 2592 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 + 144 T^{2} + 10368 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_4\times C_2$ | \( 1 + 240 T^{2} + 28800 T^{4} + 240 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.945536706990464478146177546286, −7.87610034453133981741459986903, −7.82093062013373173431685175971, −7.58420339354765717340524314315, −7.26772986397110680071738196151, −7.17152183556223483653360433100, −6.39200259162056675645166996625, −6.37675623192398542502351789208, −6.20494577969483761611031721415, −5.89480714158431894335098077380, −5.66146869234467549316520572774, −5.42634059451941202387515861568, −5.37123198709216509579861239908, −5.12989219918026085971034929763, −4.54746904003177883314268759710, −3.89467172415305145333909008026, −3.78752791798539252294808926140, −3.69171182682501508444853310717, −2.85810297230598775477696347360, −2.73370984822445473800108118628, −2.54091915502585696429376100927, −2.49852174616488983608155380727, −1.96742849332664588346930030476, −1.42251241089855163724047909662, −0.31759997894220413338792641919,
0.31759997894220413338792641919, 1.42251241089855163724047909662, 1.96742849332664588346930030476, 2.49852174616488983608155380727, 2.54091915502585696429376100927, 2.73370984822445473800108118628, 2.85810297230598775477696347360, 3.69171182682501508444853310717, 3.78752791798539252294808926140, 3.89467172415305145333909008026, 4.54746904003177883314268759710, 5.12989219918026085971034929763, 5.37123198709216509579861239908, 5.42634059451941202387515861568, 5.66146869234467549316520572774, 5.89480714158431894335098077380, 6.20494577969483761611031721415, 6.37675623192398542502351789208, 6.39200259162056675645166996625, 7.17152183556223483653360433100, 7.26772986397110680071738196151, 7.58420339354765717340524314315, 7.82093062013373173431685175971, 7.87610034453133981741459986903, 7.945536706990464478146177546286
