| L(s) = 1 | − 2.62·2-s + 3.03·3-s + 4.90·4-s + 5-s − 7.98·6-s − 4.78·7-s − 7.61·8-s + 6.23·9-s − 2.62·10-s + 2.42·11-s + 14.8·12-s − 0.534·13-s + 12.5·14-s + 3.03·15-s + 10.2·16-s + 5.95·17-s − 16.3·18-s + 4.49·19-s + 4.90·20-s − 14.5·21-s − 6.36·22-s + 5.20·23-s − 23.1·24-s + 25-s + 1.40·26-s + 9.82·27-s − 23.4·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 1.75·3-s + 2.45·4-s + 0.447·5-s − 3.25·6-s − 1.80·7-s − 2.69·8-s + 2.07·9-s − 0.830·10-s + 0.729·11-s + 4.29·12-s − 0.148·13-s + 3.36·14-s + 0.784·15-s + 2.55·16-s + 1.44·17-s − 3.86·18-s + 1.03·19-s + 1.09·20-s − 3.17·21-s − 1.35·22-s + 1.08·23-s − 4.72·24-s + 0.200·25-s + 0.275·26-s + 1.89·27-s − 4.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.085432458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085432458\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 - 3.03T + 3T^{2} \) |
| 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 + 0.534T + 13T^{2} \) |
| 17 | \( 1 - 5.95T + 17T^{2} \) |
| 19 | \( 1 - 4.49T + 19T^{2} \) |
| 23 | \( 1 - 5.20T + 23T^{2} \) |
| 29 | \( 1 + 1.85T + 29T^{2} \) |
| 31 | \( 1 + 2.95T + 31T^{2} \) |
| 37 | \( 1 - 0.0196T + 37T^{2} \) |
| 41 | \( 1 - 9.04T + 41T^{2} \) |
| 43 | \( 1 + 7.87T + 43T^{2} \) |
| 47 | \( 1 + 0.722T + 47T^{2} \) |
| 53 | \( 1 + 0.576T + 53T^{2} \) |
| 59 | \( 1 + 6.49T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 6.90T + 71T^{2} \) |
| 73 | \( 1 + 0.265T + 73T^{2} \) |
| 79 | \( 1 - 7.91T + 79T^{2} \) |
| 89 | \( 1 - 0.759T + 89T^{2} \) |
| 97 | \( 1 + 9.11T + 97T^{2} \) |
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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
−10.52972232737581614782884150842, −9.709538559436690740826767223538, −9.394605813309883312876412722995, −8.877922619848544270852883063719, −7.65558648426519006181945570881, −7.14113437951233499112762844251, −6.10113780677371376808820750729, −3.37437184068022772967511415247, −2.85596821674010775959576209599, −1.35990381714845523480424250989,
1.35990381714845523480424250989, 2.85596821674010775959576209599, 3.37437184068022772967511415247, 6.10113780677371376808820750729, 7.14113437951233499112762844251, 7.65558648426519006181945570881, 8.877922619848544270852883063719, 9.394605813309883312876412722995, 9.709538559436690740826767223538, 10.52972232737581614782884150842
