| L(s) = 1 | + 1.90·2-s − 3-s + 1.62·4-s − 5-s − 1.90·6-s + 4.91·7-s − 0.705·8-s + 9-s − 1.90·10-s + 0.743·11-s − 1.62·12-s − 13-s + 9.37·14-s + 15-s − 4.60·16-s − 1.26·17-s + 1.90·18-s + 2.17·19-s − 1.62·20-s − 4.91·21-s + 1.41·22-s + 4.67·23-s + 0.705·24-s + 25-s − 1.90·26-s − 27-s + 8.01·28-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 0.577·3-s + 0.814·4-s − 0.447·5-s − 0.777·6-s + 1.85·7-s − 0.249·8-s + 0.333·9-s − 0.602·10-s + 0.224·11-s − 0.470·12-s − 0.277·13-s + 2.50·14-s + 0.258·15-s − 1.15·16-s − 0.306·17-s + 0.449·18-s + 0.499·19-s − 0.364·20-s − 1.07·21-s + 0.302·22-s + 0.974·23-s + 0.143·24-s + 0.200·25-s − 0.373·26-s − 0.192·27-s + 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.817839987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.817839987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 7 | \( 1 - 4.91T + 7T^{2} \) |
| 11 | \( 1 - 0.743T + 11T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 - 4.67T + 23T^{2} \) |
| 29 | \( 1 - 1.35T + 29T^{2} \) |
| 37 | \( 1 - 4.01T + 37T^{2} \) |
| 41 | \( 1 + 8.25T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 - 5.79T + 47T^{2} \) |
| 53 | \( 1 + 7.31T + 53T^{2} \) |
| 59 | \( 1 + 1.46T + 59T^{2} \) |
| 61 | \( 1 + 3.01T + 61T^{2} \) |
| 67 | \( 1 + 7.45T + 67T^{2} \) |
| 71 | \( 1 + 1.96T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 9.15T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.78943042760119628510706976700, −7.31586267157933909956921916272, −6.44079672338387434489890595054, −5.66384079019712935106787708758, −4.97754983427995635667860582262, −4.62804494835197590225128178602, −3.99822535352034950534055235439, −2.98512643570451315502841856292, −2.00375432833013105294878694958, −0.894632711690210797986866354239,
0.894632711690210797986866354239, 2.00375432833013105294878694958, 2.98512643570451315502841856292, 3.99822535352034950534055235439, 4.62804494835197590225128178602, 4.97754983427995635667860582262, 5.66384079019712935106787708758, 6.44079672338387434489890595054, 7.31586267157933909956921916272, 7.78943042760119628510706976700
