| L(s) = 1 | + 23.7·2-s + 50.3·4-s + 764.·5-s + 4.07e3·7-s − 1.09e4·8-s + 1.81e4·10-s − 1.18e4·11-s + 1.08e5·13-s + 9.67e4·14-s − 2.85e5·16-s − 1.90e5·17-s + 7.64e5·19-s + 3.84e4·20-s − 2.81e5·22-s + 9.23e5·23-s − 1.36e6·25-s + 2.58e6·26-s + 2.05e5·28-s + 3.16e6·29-s − 1.00e7·31-s − 1.16e6·32-s − 4.52e6·34-s + 3.11e6·35-s + 6.60e6·37-s + 1.81e7·38-s − 8.36e6·40-s + 2.24e7·41-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.0983·4-s + 0.546·5-s + 0.642·7-s − 0.944·8-s + 0.573·10-s − 0.244·11-s + 1.05·13-s + 0.673·14-s − 1.08·16-s − 0.553·17-s + 1.34·19-s + 0.0537·20-s − 0.256·22-s + 0.687·23-s − 0.701·25-s + 1.10·26-s + 0.0631·28-s + 0.832·29-s − 1.95·31-s − 0.196·32-s − 0.580·34-s + 0.351·35-s + 0.579·37-s + 1.40·38-s − 0.516·40-s + 1.23·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.509804511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.509804511\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 3.41e6T \) |
| good | 2 | \( 1 - 23.7T + 512T^{2} \) |
| 5 | \( 1 - 764.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.07e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.18e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.08e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.90e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.64e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.23e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.16e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.00e7T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.60e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.24e7T + 3.27e14T^{2} \) |
| 47 | \( 1 + 1.96e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.18e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.05e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.39e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.09e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.75e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.38e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.12e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.98e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.15e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.93e8T + 7.60e17T^{2} \) |
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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
−9.713493667897286540168380099531, −8.963292153074662575978112608952, −7.954869370672368704185859522138, −6.71495316404548266239454848547, −5.70348652900157206453071837915, −5.12752905618112462599789619321, −4.06548513866137358410511423400, −3.13965321812144236898654749127, −1.96608533278865360170963367425, −0.77045784391427960126876018405,
0.77045784391427960126876018405, 1.96608533278865360170963367425, 3.13965321812144236898654749127, 4.06548513866137358410511423400, 5.12752905618112462599789619321, 5.70348652900157206453071837915, 6.71495316404548266239454848547, 7.954869370672368704185859522138, 8.963292153074662575978112608952, 9.713493667897286540168380099531
