| L(s) = 1 | − 2-s − 4-s + 5-s − 7-s + 3·8-s − 10-s − 4·13-s + 14-s − 16-s + 4·17-s − 4·19-s − 20-s + 6·23-s + 25-s + 4·26-s + 28-s − 4·29-s − 2·31-s − 5·32-s − 4·34-s − 35-s − 6·37-s + 4·38-s + 3·40-s − 6·41-s − 4·43-s − 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s + 1.06·8-s − 0.316·10-s − 1.10·13-s + 0.267·14-s − 1/4·16-s + 0.970·17-s − 0.917·19-s − 0.223·20-s + 1.25·23-s + 1/5·25-s + 0.784·26-s + 0.188·28-s − 0.742·29-s − 0.359·31-s − 0.883·32-s − 0.685·34-s − 0.169·35-s − 0.986·37-s + 0.648·38-s + 0.474·40-s − 0.937·41-s − 0.609·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.11161042309996, −14.53486238200727, −14.08363759743817, −13.53559640479046, −13.02797614509203, −12.42772135315754, −12.22607401257648, −11.19950673655676, −10.72358316193928, −10.23763489478188, −9.708427320252007, −9.342115464234255, −8.792104243358749, −8.279431554157252, −7.535070102493148, −7.178839096124521, −6.532992818127792, −5.746029574969314, −5.054868412810131, −4.823600076497967, −3.816991905783986, −3.310775521641324, −2.368033883082647, −1.732718543869249, −0.8352917108329686, 0,
0.8352917108329686, 1.732718543869249, 2.368033883082647, 3.310775521641324, 3.816991905783986, 4.823600076497967, 5.054868412810131, 5.746029574969314, 6.532992818127792, 7.178839096124521, 7.535070102493148, 8.279431554157252, 8.792104243358749, 9.342115464234255, 9.708427320252007, 10.23763489478188, 10.72358316193928, 11.19950673655676, 12.22607401257648, 12.42772135315754, 13.02797614509203, 13.53559640479046, 14.08363759743817, 14.53486238200727, 15.11161042309996
