| L(s) = 1 | + 4·2-s + 19.0·3-s + 16·4-s − 7.20·5-s + 76.2·6-s − 53.6·7-s + 64·8-s + 120.·9-s − 28.8·10-s + 239.·11-s + 305.·12-s − 169·13-s − 214.·14-s − 137.·15-s + 256·16-s − 1.97e3·17-s + 482.·18-s − 373.·19-s − 115.·20-s − 1.02e3·21-s + 958.·22-s + 51.4·23-s + 1.22e3·24-s − 3.07e3·25-s − 676·26-s − 2.33e3·27-s − 857.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.22·3-s + 0.5·4-s − 0.128·5-s + 0.864·6-s − 0.413·7-s + 0.353·8-s + 0.496·9-s − 0.0911·10-s + 0.597·11-s + 0.611·12-s − 0.277·13-s − 0.292·14-s − 0.157·15-s + 0.250·16-s − 1.65·17-s + 0.350·18-s − 0.237·19-s − 0.0644·20-s − 0.505·21-s + 0.422·22-s + 0.0202·23-s + 0.432·24-s − 0.983·25-s − 0.196·26-s − 0.616·27-s − 0.206·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.774880167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.774880167\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 13 | \( 1 + 169T \) |
| good | 3 | \( 1 - 19.0T + 243T^{2} \) |
| 5 | \( 1 + 7.20T + 3.12e3T^{2} \) |
| 7 | \( 1 + 53.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 239.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.97e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 373.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 51.4T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.25e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.15e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.86e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.31e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.06e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.02e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.35e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.11e4T + 8.58e9T^{2} \) |
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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
−15.95103027360583875772429883497, −14.96625483007129307391014044730, −13.94072404089799965739505584817, −12.99139502186376200844389745737, −11.46156167152171366789830604601, −9.610055249400192744031544780096, −8.229247391209956270999781472236, −6.54440263614700288038264891485, −4.18187615851047687427026452248, −2.55064058072973037014890620831,
2.55064058072973037014890620831, 4.18187615851047687427026452248, 6.54440263614700288038264891485, 8.229247391209956270999781472236, 9.610055249400192744031544780096, 11.46156167152171366789830604601, 12.99139502186376200844389745737, 13.94072404089799965739505584817, 14.96625483007129307391014044730, 15.95103027360583875772429883497
