| L(s) = 1 | − 784.·3-s + 1.19e4·5-s + 2.11e4·7-s + 4.38e5·9-s − 5.61e5·11-s − 1.10e6·13-s − 9.33e6·15-s + 5.37e6·17-s + 8.72e6·19-s − 1.65e7·21-s − 2.29e7·23-s + 9.29e7·25-s − 2.04e8·27-s + 1.74e8·29-s + 2.58e8·31-s + 4.40e8·33-s + 2.51e8·35-s − 3.35e8·37-s + 8.68e8·39-s + 4.92e8·41-s + 7.74e8·43-s + 5.21e9·45-s − 1.19e9·47-s − 1.53e9·49-s − 4.21e9·51-s − 1.79e9·53-s − 6.68e9·55-s + ⋯ |
| L(s) = 1 | − 1.86·3-s + 1.70·5-s + 0.475·7-s + 2.47·9-s − 1.05·11-s − 0.827·13-s − 3.17·15-s + 0.917·17-s + 0.808·19-s − 0.886·21-s − 0.742·23-s + 1.90·25-s − 2.74·27-s + 1.57·29-s + 1.62·31-s + 1.95·33-s + 0.810·35-s − 0.795·37-s + 1.54·39-s + 0.663·41-s + 0.803·43-s + 4.21·45-s − 0.763·47-s − 0.773·49-s − 1.70·51-s − 0.591·53-s − 1.78·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.785449845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.785449845\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 784.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 1.19e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.11e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 5.61e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.10e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.37e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 8.72e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.29e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.74e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.58e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 3.35e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 4.92e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 7.74e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.19e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.79e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 9.05e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 9.99e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.13e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.40e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 5.92e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.80e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.45e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.73e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.35e11T + 7.15e21T^{2} \) |
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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
−10.00415493602394800984919257680, −9.851772806256024382629474002494, −7.964937427750196591029580762693, −6.78785860907250990512025937141, −5.94981869893938509527314213788, −5.25087650768138650103878403950, −4.74145381938988460646316186884, −2.63314072112796378806592288887, −1.49431980557021181388184162026, −0.66429574693943635845046512276,
0.66429574693943635845046512276, 1.49431980557021181388184162026, 2.63314072112796378806592288887, 4.74145381938988460646316186884, 5.25087650768138650103878403950, 5.94981869893938509527314213788, 6.78785860907250990512025937141, 7.964937427750196591029580762693, 9.851772806256024382629474002494, 10.00415493602394800984919257680
