| L(s) = 1 | + 42.9·2-s + 45.1·3-s + 1.32e3·4-s + 608.·5-s + 1.93e3·6-s − 521.·7-s + 3.50e4·8-s − 1.76e4·9-s + 2.61e4·10-s + 6.18e4·11-s + 5.99e4·12-s + 1.57e5·13-s − 2.23e4·14-s + 2.74e4·15-s + 8.22e5·16-s − 7.57e5·18-s − 3.27e5·19-s + 8.08e5·20-s − 2.35e4·21-s + 2.65e6·22-s + 1.00e6·23-s + 1.58e6·24-s − 1.58e6·25-s + 6.77e6·26-s − 1.68e6·27-s − 6.92e5·28-s + 6.67e6·29-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.321·3-s + 2.59·4-s + 0.435·5-s + 0.609·6-s − 0.0820·7-s + 3.02·8-s − 0.896·9-s + 0.825·10-s + 1.27·11-s + 0.834·12-s + 1.53·13-s − 0.155·14-s + 0.140·15-s + 3.13·16-s − 1.70·18-s − 0.575·19-s + 1.13·20-s − 0.0263·21-s + 2.41·22-s + 0.747·23-s + 0.972·24-s − 0.810·25-s + 2.90·26-s − 0.609·27-s − 0.212·28-s + 1.75·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(11.67319838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.67319838\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 42.9T + 512T^{2} \) |
| 3 | \( 1 - 45.1T + 1.96e4T^{2} \) |
| 5 | \( 1 - 608.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 521.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.18e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.57e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 3.27e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.00e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.67e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.88e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.90e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.28e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.52e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.13e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.97e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.13e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.17e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 8.46e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.63e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.63e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.16e8T + 7.60e17T^{2} \) |
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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
−10.77848747008929900456996165985, −9.251204460426017098597912949096, −8.182120376449703811526789988538, −6.68541716686100471131338055111, −6.18861176300804687914434519755, −5.29453944423653188969237266483, −4.01920074818921902427270817263, −3.40025383543336344614577045857, −2.29824153441885054590022281075, −1.25083884508154053422111071527,
1.25083884508154053422111071527, 2.29824153441885054590022281075, 3.40025383543336344614577045857, 4.01920074818921902427270817263, 5.29453944423653188969237266483, 6.18861176300804687914434519755, 6.68541716686100471131338055111, 8.182120376449703811526789988538, 9.251204460426017098597912949096, 10.77848747008929900456996165985
