| L(s) = 1 | − 9·3-s + 21.5·5-s + 125.·7-s + 81·9-s + 251.·11-s − 1.04e3·13-s − 194.·15-s + 1.73e3·17-s + 952.·19-s − 1.12e3·21-s + 529·23-s − 2.65e3·25-s − 729·27-s + 8.87e3·29-s − 5.01e3·31-s − 2.26e3·33-s + 2.69e3·35-s − 3.17e3·37-s + 9.44e3·39-s + 2.05e4·41-s − 8.59e3·43-s + 1.74e3·45-s − 5.68e3·47-s − 1.15e3·49-s − 1.55e4·51-s + 2.21e4·53-s + 5.42e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.385·5-s + 0.964·7-s + 0.333·9-s + 0.626·11-s − 1.72·13-s − 0.222·15-s + 1.45·17-s + 0.605·19-s − 0.557·21-s + 0.208·23-s − 0.851·25-s − 0.192·27-s + 1.95·29-s − 0.937·31-s − 0.361·33-s + 0.372·35-s − 0.381·37-s + 0.994·39-s + 1.90·41-s − 0.709·43-s + 0.128·45-s − 0.375·47-s − 0.0687·49-s − 0.839·51-s + 1.08·53-s + 0.241·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.418774398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.418774398\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 - 529T \) |
| good | 5 | \( 1 - 21.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 125.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 251.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.73e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 952.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 8.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.17e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.59e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.68e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.21e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 541.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.86e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.03e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.54e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.11e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.60e4T + 8.58e9T^{2} \) |
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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
−9.370322905266025662096127227642, −8.117979937955016503673282681766, −7.48765346574386285998831707283, −6.63544761846534393102334784712, −5.49364953156070956382804524164, −5.06418625429932801410072874907, −4.05741679845997165970363373848, −2.72944614747831775202965689306, −1.63243880934314810568806707326, −0.72073294991154377265598624081,
0.72073294991154377265598624081, 1.63243880934314810568806707326, 2.72944614747831775202965689306, 4.05741679845997165970363373848, 5.06418625429932801410072874907, 5.49364953156070956382804524164, 6.63544761846534393102334784712, 7.48765346574386285998831707283, 8.117979937955016503673282681766, 9.370322905266025662096127227642
