| L(s) = 1 | − 2-s − 4-s + 3·8-s − 3·9-s − 4·11-s + 2·13-s − 16-s − 4·17-s + 3·18-s − 19-s + 4·22-s − 6·23-s − 2·26-s − 6·29-s + 4·31-s − 5·32-s + 4·34-s + 3·36-s − 10·37-s + 38-s + 10·41-s + 2·43-s + 4·44-s + 6·46-s + 6·47-s − 2·52-s + 10·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 9-s − 1.20·11-s + 0.554·13-s − 1/4·16-s − 0.970·17-s + 0.707·18-s − 0.229·19-s + 0.852·22-s − 1.25·23-s − 0.392·26-s − 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.685·34-s + 1/2·36-s − 1.64·37-s + 0.162·38-s + 1.56·41-s + 0.304·43-s + 0.603·44-s + 0.884·46-s + 0.875·47-s − 0.277·52-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.80927037546874, −15.34469056627237, −14.59920217656141, −14.00353056475203, −13.55577503339657, −13.23198965962026, −12.47137782871371, −11.92897050442017, −11.09315478279226, −10.72903292838708, −10.35626718664521, −9.513586823604723, −9.100133175352587, −8.438119017371597, −8.160651458844081, −7.540779182409274, −6.853301141604641, −5.931613350329385, −5.558072484862187, −4.840219553666075, −4.106521831421743, −3.512497882161801, −2.451671303288168, −2.002941568472231, −0.7287804435836824, 0,
0.7287804435836824, 2.002941568472231, 2.451671303288168, 3.512497882161801, 4.106521831421743, 4.840219553666075, 5.558072484862187, 5.931613350329385, 6.853301141604641, 7.540779182409274, 8.160651458844081, 8.438119017371597, 9.100133175352587, 9.513586823604723, 10.35626718664521, 10.72903292838708, 11.09315478279226, 11.92897050442017, 12.47137782871371, 13.23198965962026, 13.55577503339657, 14.00353056475203, 14.59920217656141, 15.34469056627237, 15.80927037546874
