| L(s) = 1 | − 2-s + 4-s + 3.46·5-s − 3.46·7-s − 8-s − 3.46·10-s − 3.46·13-s + 3.46·14-s + 16-s − 6·17-s + 6.92·19-s + 3.46·20-s − 3.46·23-s + 6.99·25-s + 3.46·26-s − 3.46·28-s + 6·29-s − 8·31-s − 32-s + 6·34-s − 11.9·35-s − 2·37-s − 6.92·38-s − 3.46·40-s − 6·41-s + 3.46·46-s − 10.3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.54·5-s − 1.30·7-s − 0.353·8-s − 1.09·10-s − 0.960·13-s + 0.925·14-s + 0.250·16-s − 1.45·17-s + 1.58·19-s + 0.774·20-s − 0.722·23-s + 1.39·25-s + 0.679·26-s − 0.654·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 2.02·35-s − 0.328·37-s − 1.12·38-s − 0.547·40-s − 0.937·41-s + 0.510·46-s − 1.51·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 3.46T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.045988152678817840719674687896, −7.974118909830113277898075007181, −6.84570371632624073085198253387, −6.58787292885126702103827837214, −5.67920969573548208024374822969, −4.89405740597908458181000854011, −3.35902641318813610654946834557, −2.53889350872144173817929234141, −1.64980167632186299370178743758, 0,
1.64980167632186299370178743758, 2.53889350872144173817929234141, 3.35902641318813610654946834557, 4.89405740597908458181000854011, 5.67920969573548208024374822969, 6.58787292885126702103827837214, 6.84570371632624073085198253387, 7.974118909830113277898075007181, 9.045988152678817840719674687896
