| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s − 4·10-s − 5·11-s − 13-s − 4·16-s + 17-s − 6·19-s + 4·20-s + 10·22-s + 4·23-s − 25-s + 2·26-s − 8·29-s + 31-s + 8·32-s − 2·34-s + 37-s + 12·38-s + 3·41-s + 11·43-s − 10·44-s − 8·46-s − 2·47-s + 2·50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s − 1.26·10-s − 1.50·11-s − 0.277·13-s − 16-s + 0.242·17-s − 1.37·19-s + 0.894·20-s + 2.13·22-s + 0.834·23-s − 1/5·25-s + 0.392·26-s − 1.48·29-s + 0.179·31-s + 1.41·32-s − 0.342·34-s + 0.164·37-s + 1.94·38-s + 0.468·41-s + 1.67·43-s − 1.50·44-s − 1.17·46-s − 0.291·47-s + 0.282·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5051481412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5051481412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.68456268469901, −13.14867436935761, −12.91683076865858, −12.44966127537796, −11.46335707765627, −11.14229559236737, −10.60270466496379, −10.25172942101668, −9.866506420517052, −9.202648299527852, −9.025755170619749, −8.236979114603244, −7.959753268259698, −7.300654449447281, −7.001578800541453, −6.172492269393290, −5.690717662106299, −5.211962500472691, −4.498726136675276, −3.906950065731506, −2.818416510863091, −2.408517233193856, −1.928543348430455, −1.177396749153840, −0.2946724267712450,
0.2946724267712450, 1.177396749153840, 1.928543348430455, 2.408517233193856, 2.818416510863091, 3.906950065731506, 4.498726136675276, 5.211962500472691, 5.690717662106299, 6.172492269393290, 7.001578800541453, 7.300654449447281, 7.959753268259698, 8.236979114603244, 9.025755170619749, 9.202648299527852, 9.866506420517052, 10.25172942101668, 10.60270466496379, 11.14229559236737, 11.46335707765627, 12.44966127537796, 12.91683076865858, 13.14867436935761, 13.68456268469901
