L(s) = 1 | + 2-s + 2.57·3-s + 4-s + 0.809·5-s + 2.57·6-s − 0.402·7-s + 8-s + 3.61·9-s + 0.809·10-s + 5.26·11-s + 2.57·12-s − 2.39·13-s − 0.402·14-s + 2.08·15-s + 16-s + 1.22·17-s + 3.61·18-s + 19-s + 0.809·20-s − 1.03·21-s + 5.26·22-s − 4.99·23-s + 2.57·24-s − 4.34·25-s − 2.39·26-s + 1.59·27-s − 0.402·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.48·3-s + 0.5·4-s + 0.361·5-s + 1.05·6-s − 0.152·7-s + 0.353·8-s + 1.20·9-s + 0.255·10-s + 1.58·11-s + 0.742·12-s − 0.663·13-s − 0.107·14-s + 0.537·15-s + 0.250·16-s + 0.296·17-s + 0.853·18-s + 0.229·19-s + 0.180·20-s − 0.225·21-s + 1.12·22-s − 1.04·23-s + 0.525·24-s − 0.868·25-s − 0.468·26-s + 0.306·27-s − 0.0760·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.926702744\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.926702744\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 2.57T + 3T^{2} \) |
| 5 | \( 1 - 0.809T + 5T^{2} \) |
| 7 | \( 1 + 0.402T + 7T^{2} \) |
| 11 | \( 1 - 5.26T + 11T^{2} \) |
| 13 | \( 1 + 2.39T + 13T^{2} \) |
| 17 | \( 1 - 1.22T + 17T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 - 3.55T + 31T^{2} \) |
| 37 | \( 1 - 8.84T + 37T^{2} \) |
| 41 | \( 1 - 7.57T + 41T^{2} \) |
| 43 | \( 1 - 9.17T + 43T^{2} \) |
| 47 | \( 1 + 9.62T + 47T^{2} \) |
| 53 | \( 1 + 7.00T + 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 + 1.25T + 61T^{2} \) |
| 67 | \( 1 + 6.58T + 67T^{2} \) |
| 71 | \( 1 + 5.75T + 71T^{2} \) |
| 73 | \( 1 - 1.63T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 7.81T + 83T^{2} \) |
| 89 | \( 1 - 4.35T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−7.82490372627497230309518299493, −7.24507795574260298878303495376, −6.19942823049555389290752891230, −6.09330219754862852575763100882, −4.62950633446869830202669160897, −4.27701586735323156825694068209, −3.41703419205017783451400202571, −2.79880128979375860364429019090, −2.06738054536607696967316320691, −1.18810125003194054619416186606,
1.18810125003194054619416186606, 2.06738054536607696967316320691, 2.79880128979375860364429019090, 3.41703419205017783451400202571, 4.27701586735323156825694068209, 4.62950633446869830202669160897, 6.09330219754862852575763100882, 6.19942823049555389290752891230, 7.24507795574260298878303495376, 7.82490372627497230309518299493