| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 5·11-s − 14-s + 16-s − 3·17-s − 19-s − 5·22-s + 3·23-s − 28-s − 2·29-s + 4·31-s + 32-s − 3·34-s + 4·37-s − 38-s − 7·41-s − 2·43-s − 5·44-s + 3·46-s + 6·47-s − 6·49-s + 5·53-s − 56-s − 2·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.50·11-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.229·19-s − 1.06·22-s + 0.625·23-s − 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.514·34-s + 0.657·37-s − 0.162·38-s − 1.09·41-s − 0.304·43-s − 0.753·44-s + 0.442·46-s + 0.875·47-s − 6/7·49-s + 0.686·53-s − 0.133·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.69285906093429, −13.93534918765781, −13.55381970427406, −13.03076090508294, −12.86006383321251, −12.16530416696091, −11.63915093967244, −10.96598444224250, −10.74411322701708, −10.01044656091262, −9.692212838225066, −8.800027173593455, −8.398638466497186, −7.716251755357042, −7.280450202383031, −6.563169675611916, −6.227020791967123, −5.388802189486729, −5.058729312594605, −4.481907494921072, −3.732363725676323, −3.150146113644816, −2.473005411945444, −2.076245460849143, −0.9284098938859490, 0,
0.9284098938859490, 2.076245460849143, 2.473005411945444, 3.150146113644816, 3.732363725676323, 4.481907494921072, 5.058729312594605, 5.388802189486729, 6.227020791967123, 6.563169675611916, 7.280450202383031, 7.716251755357042, 8.398638466497186, 8.800027173593455, 9.692212838225066, 10.01044656091262, 10.74411322701708, 10.96598444224250, 11.63915093967244, 12.16530416696091, 12.86006383321251, 13.03076090508294, 13.55381970427406, 13.93534918765781, 14.69285906093429
