| L(s) = 1 | − 40.3·2-s + 243·3-s − 415.·4-s + 1.36e4·5-s − 9.81e3·6-s + 8.10e4·7-s + 9.95e4·8-s + 5.90e4·9-s − 5.53e5·10-s + 2.43e5·11-s − 1.01e5·12-s − 2.52e6·13-s − 3.27e6·14-s + 3.32e6·15-s − 3.16e6·16-s − 6.01e6·17-s − 2.38e6·18-s + 1.38e7·19-s − 5.69e6·20-s + 1.97e7·21-s − 9.83e6·22-s + 3.48e7·23-s + 2.41e7·24-s + 1.38e8·25-s + 1.01e8·26-s + 1.43e7·27-s − 3.37e7·28-s + ⋯ |
| L(s) = 1 | − 0.892·2-s + 0.577·3-s − 0.203·4-s + 1.96·5-s − 0.515·6-s + 1.82·7-s + 1.07·8-s + 0.333·9-s − 1.75·10-s + 0.455·11-s − 0.117·12-s − 1.88·13-s − 1.62·14-s + 1.13·15-s − 0.755·16-s − 1.02·17-s − 0.297·18-s + 1.28·19-s − 0.398·20-s + 1.05·21-s − 0.406·22-s + 1.12·23-s + 0.620·24-s + 2.84·25-s + 1.68·26-s + 0.192·27-s − 0.370·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.255000048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.255000048\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 + 40.3T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.36e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 8.10e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.43e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.52e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.01e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.38e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.48e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 6.14e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 3.78e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 1.90e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 6.76e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.86e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.03e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.43e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 5.11e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.62e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.34e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.98e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.88e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 8.01e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 5.58e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 6.19e10T + 7.15e21T^{2} \) |
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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
−10.27435978907827799547021809756, −9.453815283868730881559070940014, −8.971432217796393756643550191394, −7.85072870214410109167166811720, −6.90457671491814486214505031997, −5.10433694394710662248395301366, −4.79759372101172297823241085047, −2.50638145172492918271485315619, −1.76481032591698096269380010868, −1.00724131998726806325373498528,
1.00724131998726806325373498528, 1.76481032591698096269380010868, 2.50638145172492918271485315619, 4.79759372101172297823241085047, 5.10433694394710662248395301366, 6.90457671491814486214505031997, 7.85072870214410109167166811720, 8.971432217796393756643550191394, 9.453815283868730881559070940014, 10.27435978907827799547021809756
