| L(s) = 1 | − 2·4-s − 5-s − 7-s − 3·9-s + 2·11-s − 5·13-s + 4·16-s + 4·19-s + 2·20-s + 5·23-s + 25-s + 2·28-s − 6·29-s − 3·31-s + 35-s + 6·36-s − 7·37-s + 3·41-s + 6·43-s − 4·44-s + 3·45-s + 47-s + 49-s + 10·52-s + 4·53-s − 2·55-s − 6·59-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s − 0.377·7-s − 9-s + 0.603·11-s − 1.38·13-s + 16-s + 0.917·19-s + 0.447·20-s + 1.04·23-s + 1/5·25-s + 0.377·28-s − 1.11·29-s − 0.538·31-s + 0.169·35-s + 36-s − 1.15·37-s + 0.468·41-s + 0.914·43-s − 0.603·44-s + 0.447·45-s + 0.145·47-s + 1/7·49-s + 1.38·52-s + 0.549·53-s − 0.269·55-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.98741047912795, −16.60044934099381, −15.68550781332412, −15.08135163945752, −14.45952481165432, −14.19984721554867, −13.54681475961765, −12.74001896571875, −12.40820603812382, −11.74128356601419, −11.15719515330872, −10.44420695235858, −9.579885264422997, −9.274084558732075, −8.771810698037885, −7.947974244601669, −7.383084807687323, −6.755662993246641, −5.677655459131873, −5.288856833975137, −4.598975722942294, −3.693356115614382, −3.230756737188618, −2.274539936551808, −0.8941623197314479, 0,
0.8941623197314479, 2.274539936551808, 3.230756737188618, 3.693356115614382, 4.598975722942294, 5.288856833975137, 5.677655459131873, 6.755662993246641, 7.383084807687323, 7.947974244601669, 8.771810698037885, 9.274084558732075, 9.579885264422997, 10.44420695235858, 11.15719515330872, 11.74128356601419, 12.40820603812382, 12.74001896571875, 13.54681475961765, 14.19984721554867, 14.45952481165432, 15.08135163945752, 15.68550781332412, 16.60044934099381, 16.98741047912795
