| L(s) = 1 | + 0.241·2-s + 0.709·3-s − 0.941·4-s + 5-s + 0.170·6-s + 1.77·7-s − 0.468·8-s − 0.497·9-s + 0.241·10-s − 1.13·11-s − 0.667·12-s − 13-s + 0.426·14-s + 0.709·15-s + 0.829·16-s + 1.94·17-s − 0.119·18-s − 0.941·20-s + 1.25·21-s − 0.273·22-s + 1.49·23-s − 0.332·24-s + 25-s − 0.241·26-s − 1.06·27-s − 1.66·28-s + 0.170·30-s + ⋯ |
| L(s) = 1 | + 0.241·2-s + 0.709·3-s − 0.941·4-s + 5-s + 0.170·6-s + 1.77·7-s − 0.468·8-s − 0.497·9-s + 0.241·10-s − 1.13·11-s − 0.667·12-s − 13-s + 0.426·14-s + 0.709·15-s + 0.829·16-s + 1.94·17-s − 0.119·18-s − 0.941·20-s + 1.25·21-s − 0.273·22-s + 1.49·23-s − 0.332·24-s + 25-s − 0.241·26-s − 1.06·27-s − 1.66·28-s + 0.170·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.714477308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.714477308\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.241T + T^{2} \) |
| 3 | \( 1 - 0.709T + T^{2} \) |
| 7 | \( 1 - 1.77T + T^{2} \) |
| 11 | \( 1 + 1.13T + T^{2} \) |
| 17 | \( 1 - 1.94T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.49T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.241T + T^{2} \) |
| 47 | \( 1 + 1.49T + T^{2} \) |
| 53 | \( 1 + 1.13T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.13T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.77T + T^{2} \) |
| 97 | \( 1 + 0.709T + T^{2} \) |
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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
−9.345820184685168468928134692734, −8.439442584270301628705003584723, −8.016504757606841357863568687558, −7.31393206007081334580384801897, −5.68399297768453450262517834071, −5.19236727407332585456642931927, −4.83317905439022467962790964152, −3.36335821906628261504053968807, −2.57045509104693024712292247872, −1.43062094492485354654680844975,
1.43062094492485354654680844975, 2.57045509104693024712292247872, 3.36335821906628261504053968807, 4.83317905439022467962790964152, 5.19236727407332585456642931927, 5.68399297768453450262517834071, 7.31393206007081334580384801897, 8.016504757606841357863568687558, 8.439442584270301628705003584723, 9.345820184685168468928134692734
