| L(s) = 1 | − 2.49·3-s + 2.79·5-s − 1.31·7-s + 3.23·9-s − 0.214·11-s + 4.48·13-s − 6.98·15-s + 0.531·17-s + 7.59·19-s + 3.28·21-s + 6.80·23-s + 2.83·25-s − 0.583·27-s + 10.5·29-s − 3.24·31-s + 0.536·33-s − 3.68·35-s − 5.24·37-s − 11.2·39-s + 3.41·41-s + 7.59·43-s + 9.05·45-s + 5.87·47-s − 5.27·49-s − 1.32·51-s + 3.35·53-s − 0.601·55-s + ⋯ |
| L(s) = 1 | − 1.44·3-s + 1.25·5-s − 0.496·7-s + 1.07·9-s − 0.0647·11-s + 1.24·13-s − 1.80·15-s + 0.128·17-s + 1.74·19-s + 0.716·21-s + 1.41·23-s + 0.567·25-s − 0.112·27-s + 1.96·29-s − 0.582·31-s + 0.0934·33-s − 0.622·35-s − 0.861·37-s − 1.79·39-s + 0.533·41-s + 1.15·43-s + 1.34·45-s + 0.856·47-s − 0.753·49-s − 0.185·51-s + 0.461·53-s − 0.0811·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.870638660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.870638660\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 + 0.214T + 11T^{2} \) |
| 13 | \( 1 - 4.48T + 13T^{2} \) |
| 17 | \( 1 - 0.531T + 17T^{2} \) |
| 19 | \( 1 - 7.59T + 19T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 3.24T + 31T^{2} \) |
| 37 | \( 1 + 5.24T + 37T^{2} \) |
| 41 | \( 1 - 3.41T + 41T^{2} \) |
| 43 | \( 1 - 7.59T + 43T^{2} \) |
| 47 | \( 1 - 5.87T + 47T^{2} \) |
| 53 | \( 1 - 3.35T + 53T^{2} \) |
| 59 | \( 1 - 9.93T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 3.31T + 71T^{2} \) |
| 73 | \( 1 + 2.20T + 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 - 3.88T + 83T^{2} \) |
| 89 | \( 1 + 6.38T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−7.55050794526594351151048319870, −6.89380897931332568996599073932, −6.21633893120061201590366396102, −5.83117093833995959118065683582, −5.26340590866603152378497451856, −4.62105173936224487585561687568, −3.44105502619610671144713479701, −2.70492028158683785923378308658, −1.36121258065766087240366763252, −0.845551803525540769009745422640,
0.845551803525540769009745422640, 1.36121258065766087240366763252, 2.70492028158683785923378308658, 3.44105502619610671144713479701, 4.62105173936224487585561687568, 5.26340590866603152378497451856, 5.83117093833995959118065683582, 6.21633893120061201590366396102, 6.89380897931332568996599073932, 7.55050794526594351151048319870
