| L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s − 3·11-s − 4·13-s − 14-s + 16-s − 4·19-s − 3·20-s + 3·22-s − 6·23-s + 4·25-s + 4·26-s + 28-s − 3·29-s − 31-s − 32-s − 3·35-s + 2·37-s + 4·38-s + 3·40-s + 2·43-s − 3·44-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s − 0.904·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.670·20-s + 0.639·22-s − 1.25·23-s + 4/5·25-s + 0.784·26-s + 0.188·28-s − 0.557·29-s − 0.179·31-s − 0.176·32-s − 0.507·35-s + 0.328·37-s + 0.648·38-s + 0.474·40-s + 0.304·43-s − 0.452·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−15.11639969211959, −14.91928504869725, −14.32442435977080, −13.56566675722087, −12.90142083493516, −12.36523035691539, −11.90931598250853, −11.54698582081689, −10.77733187409575, −10.53854269070572, −9.885698990185076, −9.248997050939805, −8.617265875762957, −7.989254903091512, −7.777820429621939, −7.303415427343906, −6.633319582959625, −5.859342469787674, −5.190145690009507, −4.487213771642666, −3.989713683590020, −3.231937661028310, −2.386057355199235, −1.935011619914818, −0.6380976009538475, 0,
0.6380976009538475, 1.935011619914818, 2.386057355199235, 3.231937661028310, 3.989713683590020, 4.487213771642666, 5.190145690009507, 5.859342469787674, 6.633319582959625, 7.303415427343906, 7.777820429621939, 7.989254903091512, 8.617265875762957, 9.248997050939805, 9.885698990185076, 10.53854269070572, 10.77733187409575, 11.54698582081689, 11.90931598250853, 12.36523035691539, 12.90142083493516, 13.56566675722087, 14.32442435977080, 14.91928504869725, 15.11639969211959
