| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s + 4·11-s − 12-s − 6·13-s + 2·15-s + 16-s + 18-s + 4·19-s − 2·20-s + 4·22-s − 8·23-s − 24-s − 25-s − 6·26-s − 27-s + 2·29-s + 2·30-s + 32-s − 4·33-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 1.66·13-s + 0.516·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.852·22-s − 1.66·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.371·29-s + 0.365·30-s + 0.176·32-s − 0.696·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.27640114607469, −13.69790765079136, −13.20066206236938, −12.39220018137305, −12.12998307769760, −11.82068782940482, −11.51810032687050, −10.92445318935126, −10.13604535478320, −9.773955567109087, −9.411119735119491, −8.467877472855487, −7.924513137605369, −7.489255661210195, −6.995364485348405, −6.457198061579472, −5.944641925767685, −5.258321352047645, −4.814874282641343, −4.095626606948838, −3.937364724229683, −3.135875733391663, −2.403401334294822, −1.717822133723067, −0.8321233335871864, 0,
0.8321233335871864, 1.717822133723067, 2.403401334294822, 3.135875733391663, 3.937364724229683, 4.095626606948838, 4.814874282641343, 5.258321352047645, 5.944641925767685, 6.457198061579472, 6.995364485348405, 7.489255661210195, 7.924513137605369, 8.467877472855487, 9.411119735119491, 9.773955567109087, 10.13604535478320, 10.92445318935126, 11.51810032687050, 11.82068782940482, 12.12998307769760, 12.39220018137305, 13.20066206236938, 13.69790765079136, 14.27640114607469
