| L(s) = 1 | − 2-s + 1.46·3-s + 4-s − 0.449·5-s − 1.46·6-s + 1.96·7-s − 8-s − 0.859·9-s + 0.449·10-s + 4.84·11-s + 1.46·12-s + 0.267·13-s − 1.96·14-s − 0.657·15-s + 16-s − 6.27·17-s + 0.859·18-s − 5.36·19-s − 0.449·20-s + 2.87·21-s − 4.84·22-s − 2.72·23-s − 1.46·24-s − 4.79·25-s − 0.267·26-s − 5.64·27-s + 1.96·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.844·3-s + 0.5·4-s − 0.200·5-s − 0.597·6-s + 0.743·7-s − 0.353·8-s − 0.286·9-s + 0.142·10-s + 1.46·11-s + 0.422·12-s + 0.0741·13-s − 0.525·14-s − 0.169·15-s + 0.250·16-s − 1.52·17-s + 0.202·18-s − 1.23·19-s − 0.100·20-s + 0.627·21-s − 1.03·22-s − 0.568·23-s − 0.298·24-s − 0.959·25-s − 0.0524·26-s − 1.08·27-s + 0.371·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 1.46T + 3T^{2} \) |
| 5 | \( 1 + 0.449T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 - 4.84T + 11T^{2} \) |
| 13 | \( 1 - 0.267T + 13T^{2} \) |
| 17 | \( 1 + 6.27T + 17T^{2} \) |
| 19 | \( 1 + 5.36T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 - 5.87T + 29T^{2} \) |
| 31 | \( 1 + 2.11T + 31T^{2} \) |
| 37 | \( 1 + 4.54T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 3.18T + 43T^{2} \) |
| 47 | \( 1 - 0.522T + 47T^{2} \) |
| 53 | \( 1 + 6.53T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 6.04T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 3.42T + 73T^{2} \) |
| 79 | \( 1 - 2.19T + 79T^{2} \) |
| 83 | \( 1 - 1.59T + 83T^{2} \) |
| 89 | \( 1 - 3.48T + 89T^{2} \) |
| 97 | \( 1 - 0.151T + 97T^{2} \) |
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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
−8.331584478188889515742703849947, −7.62641047649398981962661539524, −6.55614380846689391668618789599, −6.31203550760927129153163242537, −4.92656438421267709906968908730, −4.09868316065996480354817277871, −3.36379885446803446689620197821, −2.14647869998751295723071819062, −1.68154772439674745689754835693, 0,
1.68154772439674745689754835693, 2.14647869998751295723071819062, 3.36379885446803446689620197821, 4.09868316065996480354817277871, 4.92656438421267709906968908730, 6.31203550760927129153163242537, 6.55614380846689391668618789599, 7.62641047649398981962661539524, 8.331584478188889515742703849947
