| L(s) = 1 | + 2·2-s + 4·4-s + 10.4·5-s + 4.15·7-s + 8·8-s + 20.8·10-s − 68.1·13-s + 8.31·14-s + 16·16-s − 25.5·17-s + 10.5·19-s + 41.7·20-s − 169.·23-s − 16.0·25-s − 136.·26-s + 16.6·28-s + 115.·29-s − 122.·31-s + 32·32-s − 51.1·34-s + 43.4·35-s + 272.·37-s + 21.1·38-s + 83.5·40-s − 58.6·41-s − 19.8·43-s − 338.·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.933·5-s + 0.224·7-s + 0.353·8-s + 0.660·10-s − 1.45·13-s + 0.158·14-s + 0.250·16-s − 0.364·17-s + 0.127·19-s + 0.466·20-s − 1.53·23-s − 0.128·25-s − 1.02·26-s + 0.112·28-s + 0.742·29-s − 0.709·31-s + 0.176·32-s − 0.257·34-s + 0.209·35-s + 1.21·37-s + 0.0901·38-s + 0.330·40-s − 0.223·41-s − 0.0704·43-s − 1.08·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 10.4T + 125T^{2} \) |
| 7 | \( 1 - 4.15T + 343T^{2} \) |
| 13 | \( 1 + 68.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 25.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 115.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 272.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 58.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 19.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 450.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 486.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 806.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 755.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 495.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 62.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 179.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 16.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 900.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 692.T + 9.12e5T^{2} \) |
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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
−8.092921306735398295902774228808, −7.55189415418356981919588381733, −6.46073950444206543450740662608, −6.01135898984970746691565768909, −4.99932775781094651229868429233, −4.51852674155468459301994419276, −3.28946815787533098719772770541, −2.31274797560516287369952917912, −1.66502635406597654429787846151, 0,
1.66502635406597654429787846151, 2.31274797560516287369952917912, 3.28946815787533098719772770541, 4.51852674155468459301994419276, 4.99932775781094651229868429233, 6.01135898984970746691565768909, 6.46073950444206543450740662608, 7.55189415418356981919588381733, 8.092921306735398295902774228808
