| L(s) = 1 | − 1.11·2-s − 2.03·3-s − 0.758·4-s − 1.43·5-s + 2.26·6-s − 7-s + 3.07·8-s + 1.13·9-s + 1.59·10-s + 4.29·11-s + 1.54·12-s + 3.71·13-s + 1.11·14-s + 2.91·15-s − 1.90·16-s + 5.27·17-s − 1.26·18-s − 1.78·19-s + 1.08·20-s + 2.03·21-s − 4.78·22-s + 5.71·23-s − 6.24·24-s − 2.94·25-s − 4.14·26-s + 3.79·27-s + 0.758·28-s + ⋯ |
| L(s) = 1 | − 0.787·2-s − 1.17·3-s − 0.379·4-s − 0.641·5-s + 0.924·6-s − 0.377·7-s + 1.08·8-s + 0.377·9-s + 0.505·10-s + 1.29·11-s + 0.445·12-s + 1.03·13-s + 0.297·14-s + 0.752·15-s − 0.477·16-s + 1.28·17-s − 0.297·18-s − 0.409·19-s + 0.243·20-s + 0.443·21-s − 1.02·22-s + 1.19·23-s − 1.27·24-s − 0.589·25-s − 0.812·26-s + 0.730·27-s + 0.143·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6007979207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6007979207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 2 | \( 1 + 1.11T + 2T^{2} \) |
| 3 | \( 1 + 2.03T + 3T^{2} \) |
| 5 | \( 1 + 1.43T + 5T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 13 | \( 1 - 3.71T + 13T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 19 | \( 1 + 1.78T + 19T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 - 0.667T + 29T^{2} \) |
| 31 | \( 1 - 5.78T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 - 3.76T + 41T^{2} \) |
| 43 | \( 1 + 5.75T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 - 3.49T + 59T^{2} \) |
| 61 | \( 1 - 3.08T + 61T^{2} \) |
| 67 | \( 1 - 8.22T + 67T^{2} \) |
| 71 | \( 1 - 5.27T + 71T^{2} \) |
| 73 | \( 1 + 8.00T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 4.40T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 0.803T + 97T^{2} \) |
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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
−8.615030002607708636927974469857, −8.226061605554812261573466730588, −7.15907258780135621537058535779, −6.53434865327623677878345192699, −5.76716783570696153218470474728, −4.89909346868990972013395244593, −4.05109481414770162650143744888, −3.31793919125207952122581540019, −1.37512878933220926215785709836, −0.65249792980652917422038985167,
0.65249792980652917422038985167, 1.37512878933220926215785709836, 3.31793919125207952122581540019, 4.05109481414770162650143744888, 4.89909346868990972013395244593, 5.76716783570696153218470474728, 6.53434865327623677878345192699, 7.15907258780135621537058535779, 8.226061605554812261573466730588, 8.615030002607708636927974469857
