| L(s) = 1 | + 8.23·2-s + 11.6·3-s + 35.8·4-s + 95.8·6-s − 195.·7-s + 31.9·8-s − 107.·9-s + 64.5·11-s + 417.·12-s + 169·13-s − 1.60e3·14-s − 885.·16-s + 426.·17-s − 886.·18-s − 959.·19-s − 2.27e3·21-s + 531.·22-s − 499.·23-s + 371.·24-s + 1.39e3·26-s − 4.07e3·27-s − 6.99e3·28-s + 1.28e3·29-s − 6.73e3·31-s − 8.31e3·32-s + 751.·33-s + 3.50e3·34-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 0.746·3-s + 1.12·4-s + 1.08·6-s − 1.50·7-s + 0.176·8-s − 0.442·9-s + 0.160·11-s + 0.836·12-s + 0.277·13-s − 2.19·14-s − 0.864·16-s + 0.357·17-s − 0.644·18-s − 0.609·19-s − 1.12·21-s + 0.234·22-s − 0.196·23-s + 0.131·24-s + 0.403·26-s − 1.07·27-s − 1.68·28-s + 0.284·29-s − 1.25·31-s − 1.43·32-s + 0.120·33-s + 0.520·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 8.23T + 32T^{2} \) |
| 3 | \( 1 - 11.6T + 243T^{2} \) |
| 7 | \( 1 + 195.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 64.5T + 1.61e5T^{2} \) |
| 17 | \( 1 - 426.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 959.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 499.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.21e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.49e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.56e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.03e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.56e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.91e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.45e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.81e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.33e4T + 8.58e9T^{2} \) |
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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
−10.36135116299176545805545148329, −9.303601173925032901600279594453, −8.511630872512766324982040592189, −7.02750572149212878156652535519, −6.21090166456278466689259959614, −5.30949018839157577329638233947, −3.79661444868374504347434088296, −3.32915391293092424549672969131, −2.27009398876332768304454620923, 0,
2.27009398876332768304454620923, 3.32915391293092424549672969131, 3.79661444868374504347434088296, 5.30949018839157577329638233947, 6.21090166456278466689259959614, 7.02750572149212878156652535519, 8.511630872512766324982040592189, 9.303601173925032901600279594453, 10.36135116299176545805545148329
