L(s) = 1 | − 2-s − 2.21·3-s + 4-s + 2.21·6-s + 3.90·7-s − 8-s + 1.90·9-s + 1.06·11-s − 2.21·12-s − 3.90·14-s + 16-s − 1.31·17-s − 1.90·18-s − 6.59·19-s − 8.64·21-s − 1.06·22-s − 2.14·23-s + 2.21·24-s + 2.42·27-s + 3.90·28-s − 9.05·29-s − 6.92·31-s − 32-s − 2.36·33-s + 1.31·34-s + 1.90·36-s − 5.02·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.27·3-s + 0.5·4-s + 0.903·6-s + 1.47·7-s − 0.353·8-s + 0.634·9-s + 0.321·11-s − 0.639·12-s − 1.04·14-s + 0.250·16-s − 0.317·17-s − 0.448·18-s − 1.51·19-s − 1.88·21-s − 0.227·22-s − 0.447·23-s + 0.451·24-s + 0.467·27-s + 0.737·28-s − 1.68·29-s − 1.24·31-s − 0.176·32-s − 0.411·33-s + 0.224·34-s + 0.317·36-s − 0.825·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6549305809\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6549305809\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 2.21T + 3T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 11 | \( 1 - 1.06T + 11T^{2} \) |
| 17 | \( 1 + 1.31T + 17T^{2} \) |
| 19 | \( 1 + 6.59T + 19T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 + 9.05T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 5.02T + 37T^{2} \) |
| 41 | \( 1 + 5.95T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 0.0967T + 47T^{2} \) |
| 53 | \( 1 - 1.49T + 53T^{2} \) |
| 59 | \( 1 - 7.18T + 59T^{2} \) |
| 61 | \( 1 - 7.88T + 61T^{2} \) |
| 67 | \( 1 - 8.42T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 4.30T + 79T^{2} \) |
| 83 | \( 1 + 9.69T + 83T^{2} \) |
| 89 | \( 1 - 4.52T + 89T^{2} \) |
| 97 | \( 1 - 4.26T + 97T^{2} \) |
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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
−7.79369455734010450599272965707, −7.05020600746311156549192499644, −6.50527431029781251725580115575, −5.64934626810453861848172571677, −5.23514924910074340361291161461, −4.41366953346542276983094401719, −3.65448151829046026702666446707, −2.09378404089998270756112775242, −1.70478612127725719190424065878, −0.47113225785072930267190113684,
0.47113225785072930267190113684, 1.70478612127725719190424065878, 2.09378404089998270756112775242, 3.65448151829046026702666446707, 4.41366953346542276983094401719, 5.23514924910074340361291161461, 5.64934626810453861848172571677, 6.50527431029781251725580115575, 7.05020600746311156549192499644, 7.79369455734010450599272965707