| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s − 4·7-s + 8-s + 9-s − 2·10-s − 12-s − 13-s − 4·14-s + 2·15-s + 16-s + 6·17-s + 18-s + 19-s − 2·20-s + 4·21-s − 4·23-s − 24-s − 25-s − 26-s − 27-s − 4·28-s − 29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s + 0.872·21-s − 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.755·28-s − 0.185·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42978 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42978 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−15.09688910770468, −14.43753037104208, −13.81244624664274, −13.40336215615006, −12.69027702829941, −12.39867911718720, −11.91297508678669, −11.62262297527100, −10.86151874149684, −10.26092794107348, −9.772407747869918, −9.483485465365959, −8.416604163841206, −7.873093177090433, −7.403421685502794, −6.717840387571606, −6.354512112692512, −5.648283108749873, −5.281898205399340, −4.354321818126516, −3.924777015874324, −3.267127480630090, −2.905507752455362, −1.835881932614478, −0.8034173549088122, 0,
0.8034173549088122, 1.835881932614478, 2.905507752455362, 3.267127480630090, 3.924777015874324, 4.354321818126516, 5.281898205399340, 5.648283108749873, 6.354512112692512, 6.717840387571606, 7.403421685502794, 7.873093177090433, 8.416604163841206, 9.483485465365959, 9.772407747869918, 10.26092794107348, 10.86151874149684, 11.62262297527100, 11.91297508678669, 12.39867911718720, 12.69027702829941, 13.40336215615006, 13.81244624664274, 14.43753037104208, 15.09688910770468
