| L(s) = 1 | − 2.58·2-s − 1.69·3-s + 4.68·4-s + 4.38·6-s − 2.28·7-s − 6.95·8-s − 0.129·9-s + 11-s − 7.94·12-s + 1.76·13-s + 5.91·14-s + 8.60·16-s + 7.54·17-s + 0.333·18-s − 19-s + 3.87·21-s − 2.58·22-s + 6.89·23-s + 11.7·24-s − 4.56·26-s + 5.30·27-s − 10.7·28-s + 6.12·29-s + 8.39·31-s − 8.34·32-s − 1.69·33-s − 19.5·34-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.978·3-s + 2.34·4-s + 1.78·6-s − 0.864·7-s − 2.45·8-s − 0.0430·9-s + 0.301·11-s − 2.29·12-s + 0.489·13-s + 1.58·14-s + 2.15·16-s + 1.83·17-s + 0.0787·18-s − 0.229·19-s + 0.846·21-s − 0.551·22-s + 1.43·23-s + 2.40·24-s − 0.894·26-s + 1.02·27-s − 2.02·28-s + 1.13·29-s + 1.50·31-s − 1.47·32-s − 0.294·33-s − 3.34·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5930522033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5930522033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 + 1.69T + 3T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 - 7.54T + 17T^{2} \) |
| 23 | \( 1 - 6.89T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 - 8.39T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 4.07T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 + 8.70T + 73T^{2} \) |
| 79 | \( 1 - 4.40T + 79T^{2} \) |
| 83 | \( 1 + 0.146T + 83T^{2} \) |
| 89 | \( 1 + 1.54T + 89T^{2} \) |
| 97 | \( 1 - 7.32T + 97T^{2} \) |
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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
−8.254281641584486756833128853181, −7.66770571685439936105040872873, −6.69530401333914072316302565304, −6.37470430721832913284404163632, −5.74436056248877882741328615991, −4.71150173019947864796644406494, −3.25932753888294332064947660479, −2.70857259941319415596866916510, −1.15308681206483535667537183174, −0.71538213076460783716251036149,
0.71538213076460783716251036149, 1.15308681206483535667537183174, 2.70857259941319415596866916510, 3.25932753888294332064947660479, 4.71150173019947864796644406494, 5.74436056248877882741328615991, 6.37470430721832913284404163632, 6.69530401333914072316302565304, 7.66770571685439936105040872873, 8.254281641584486756833128853181
