| L(s) = 1 | + 2.12·2-s + 2.49·4-s + 5-s + 4.22·7-s + 1.05·8-s + 2.12·10-s − 2.64·11-s + 2.74·13-s + 8.95·14-s − 2.75·16-s + 3.35·17-s − 3.90·19-s + 2.49·20-s − 5.61·22-s + 1.83·23-s + 25-s + 5.81·26-s + 10.5·28-s + 5.52·29-s + 8.29·31-s − 7.95·32-s + 7.10·34-s + 4.22·35-s + 1.73·37-s − 8.27·38-s + 1.05·40-s − 3.58·41-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.24·4-s + 0.447·5-s + 1.59·7-s + 0.372·8-s + 0.670·10-s − 0.798·11-s + 0.760·13-s + 2.39·14-s − 0.689·16-s + 0.812·17-s − 0.895·19-s + 0.558·20-s − 1.19·22-s + 0.383·23-s + 0.200·25-s + 1.14·26-s + 1.99·28-s + 1.02·29-s + 1.48·31-s − 1.40·32-s + 1.21·34-s + 0.713·35-s + 0.284·37-s − 1.34·38-s + 0.166·40-s − 0.560·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.849783228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.849783228\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 2.12T + 2T^{2} \) |
| 7 | \( 1 - 4.22T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 17 | \( 1 - 3.35T + 17T^{2} \) |
| 19 | \( 1 + 3.90T + 19T^{2} \) |
| 23 | \( 1 - 1.83T + 23T^{2} \) |
| 29 | \( 1 - 5.52T + 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 - 1.73T + 37T^{2} \) |
| 41 | \( 1 + 3.58T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 + 2.65T + 47T^{2} \) |
| 53 | \( 1 + 7.44T + 53T^{2} \) |
| 59 | \( 1 + 3.81T + 59T^{2} \) |
| 61 | \( 1 - 0.846T + 61T^{2} \) |
| 67 | \( 1 + 2.43T + 67T^{2} \) |
| 71 | \( 1 + 4.15T + 71T^{2} \) |
| 73 | \( 1 - 6.49T + 73T^{2} \) |
| 79 | \( 1 - 3.03T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.242983820992686435259008923040, −7.75248544265129925190083572115, −6.61802649579928938595328585585, −6.04140591810219227765875316302, −5.22221154401928346255622570106, −4.78947144769325775093472918399, −4.10302098875829618089088412182, −3.02501834204082638959641496609, −2.28164473966320247605128797003, −1.23079799877952231616513202147,
1.23079799877952231616513202147, 2.28164473966320247605128797003, 3.02501834204082638959641496609, 4.10302098875829618089088412182, 4.78947144769325775093472918399, 5.22221154401928346255622570106, 6.04140591810219227765875316302, 6.61802649579928938595328585585, 7.75248544265129925190083572115, 8.242983820992686435259008923040
