| L(s) = 1 | + 2-s − 0.503·3-s + 4-s − 5-s − 0.503·6-s − 4.37·7-s + 8-s − 2.74·9-s − 10-s − 4.14·11-s − 0.503·12-s + 4.25·13-s − 4.37·14-s + 0.503·15-s + 16-s − 1.89·17-s − 2.74·18-s − 3.65·19-s − 20-s + 2.20·21-s − 4.14·22-s − 3.31·23-s − 0.503·24-s + 25-s + 4.25·26-s + 2.89·27-s − 4.37·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.290·3-s + 0.5·4-s − 0.447·5-s − 0.205·6-s − 1.65·7-s + 0.353·8-s − 0.915·9-s − 0.316·10-s − 1.25·11-s − 0.145·12-s + 1.17·13-s − 1.16·14-s + 0.130·15-s + 0.250·16-s − 0.459·17-s − 0.647·18-s − 0.837·19-s − 0.223·20-s + 0.480·21-s − 0.884·22-s − 0.691·23-s − 0.102·24-s + 0.200·25-s + 0.833·26-s + 0.557·27-s − 0.826·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9279244425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9279244425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 0.503T + 3T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 4.25T + 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 23 | \( 1 + 3.31T + 23T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 + 0.405T + 31T^{2} \) |
| 37 | \( 1 + 3.77T + 37T^{2} \) |
| 41 | \( 1 - 3.67T + 41T^{2} \) |
| 43 | \( 1 + 1.67T + 43T^{2} \) |
| 47 | \( 1 + 2.95T + 47T^{2} \) |
| 53 | \( 1 - 2.50T + 53T^{2} \) |
| 59 | \( 1 - 4.43T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.79T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 9.53T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.236513806731612672723423952602, −7.03361282804848374933228155583, −6.60618339234584405437752904116, −5.77264766503596280416477290512, −5.50438565082064606401142849620, −4.29973232547522489274410336912, −3.62204169046803959970970833905, −2.97859650130794568153594778639, −2.20190781980848523580444213634, −0.42570704087292637851718071141,
0.42570704087292637851718071141, 2.20190781980848523580444213634, 2.97859650130794568153594778639, 3.62204169046803959970970833905, 4.29973232547522489274410336912, 5.50438565082064606401142849620, 5.77264766503596280416477290512, 6.60618339234584405437752904116, 7.03361282804848374933228155583, 8.236513806731612672723423952602
