| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 4.33·7-s − 8-s + 9-s + 10-s − 4.74·11-s − 12-s + 0.409·13-s − 4.33·14-s + 15-s + 16-s + 5.77·17-s − 18-s − 5.77·19-s − 20-s − 4.33·21-s + 4.74·22-s − 9.36·23-s + 24-s + 25-s − 0.409·26-s − 27-s + 4.33·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.63·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.43·11-s − 0.288·12-s + 0.113·13-s − 1.15·14-s + 0.258·15-s + 0.250·16-s + 1.40·17-s − 0.235·18-s − 1.32·19-s − 0.223·20-s − 0.946·21-s + 1.01·22-s − 1.95·23-s + 0.204·24-s + 0.200·25-s − 0.0802·26-s − 0.192·27-s + 0.819·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 4.33T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 - 0.409T + 13T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 + 5.77T + 19T^{2} \) |
| 23 | \( 1 + 9.36T + 23T^{2} \) |
| 29 | \( 1 + 1.30T + 29T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 - 1.71T + 41T^{2} \) |
| 43 | \( 1 + 9.49T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 1.71T + 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 71 | \( 1 - 8.11T + 71T^{2} \) |
| 73 | \( 1 + 8.05T + 73T^{2} \) |
| 79 | \( 1 - 7.64T + 79T^{2} \) |
| 83 | \( 1 - 8.57T + 83T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−8.312674131622042955891841841587, −8.083369029764618564055390938816, −7.60004782112254431886920527310, −6.42046436249885532765643778717, −5.52237848353146710178274141813, −4.85497104144803748572807269126, −3.88181068887422354669144568566, −2.42792012060599601742550466688, −1.46283408916974153530786996367, 0,
1.46283408916974153530786996367, 2.42792012060599601742550466688, 3.88181068887422354669144568566, 4.85497104144803748572807269126, 5.52237848353146710178274141813, 6.42046436249885532765643778717, 7.60004782112254431886920527310, 8.083369029764618564055390938816, 8.312674131622042955891841841587
