| L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 2·13-s − 14-s + 16-s − 6·17-s − 4·19-s + 2·20-s + 23-s − 25-s + 2·26-s − 28-s − 2·29-s − 8·31-s + 32-s − 6·34-s − 2·35-s + 6·37-s − 4·38-s + 2·40-s − 6·41-s + 4·43-s + 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.447·20-s + 0.208·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.371·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.338·35-s + 0.986·37-s − 0.648·38-s + 0.316·40-s − 0.937·41-s + 0.609·43-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350658 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350658 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.77080452383078, −12.64756241526426, −11.89623169046532, −11.34054891339026, −10.95844297006785, −10.70094200105185, −10.11941166975301, −9.479499116490601, −9.292573750016852, −8.671455432570419, −8.285291011994728, −7.591127135245977, −7.038183970948026, −6.615469775443001, −6.253921911946741, −5.707740587618762, −5.453861012173577, −4.690414269912738, −4.269530678811020, −3.766621969388987, −3.261803258412725, −2.485485508456682, −2.118984620494349, −1.727378424972155, −0.8303759589739566, 0,
0.8303759589739566, 1.727378424972155, 2.118984620494349, 2.485485508456682, 3.261803258412725, 3.766621969388987, 4.269530678811020, 4.690414269912738, 5.453861012173577, 5.707740587618762, 6.253921911946741, 6.615469775443001, 7.038183970948026, 7.591127135245977, 8.285291011994728, 8.671455432570419, 9.292573750016852, 9.479499116490601, 10.11941166975301, 10.70094200105185, 10.95844297006785, 11.34054891339026, 11.89623169046532, 12.64756241526426, 12.77080452383078
