| L(s) = 1 | − 1.67·3-s − 5-s + 7-s − 0.193·9-s − 11-s − 1.67·13-s + 1.67·15-s − 6.24·17-s + 3.73·19-s − 1.67·21-s + 8.73·23-s + 25-s + 5.35·27-s − 4.38·29-s − 7.44·31-s + 1.67·33-s − 35-s + 11.4·37-s + 2.80·39-s + 4.79·41-s + 10.5·43-s + 0.193·45-s − 6.89·47-s + 49-s + 10.4·51-s − 6.80·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.967·3-s − 0.447·5-s + 0.377·7-s − 0.0646·9-s − 0.301·11-s − 0.464·13-s + 0.432·15-s − 1.51·17-s + 0.857·19-s − 0.365·21-s + 1.82·23-s + 0.200·25-s + 1.02·27-s − 0.814·29-s − 1.33·31-s + 0.291·33-s − 0.169·35-s + 1.87·37-s + 0.449·39-s + 0.748·41-s + 1.60·43-s + 0.0289·45-s − 1.00·47-s + 0.142·49-s + 1.46·51-s − 0.934·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.67T + 3T^{2} \) |
| 13 | \( 1 + 1.67T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 - 8.73T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.89T + 47T^{2} \) |
| 53 | \( 1 + 6.80T + 53T^{2} \) |
| 59 | \( 1 - 3.13T + 59T^{2} \) |
| 61 | \( 1 + 4.40T + 61T^{2} \) |
| 67 | \( 1 + 2.46T + 67T^{2} \) |
| 71 | \( 1 - 8.31T + 71T^{2} \) |
| 73 | \( 1 + 4.45T + 73T^{2} \) |
| 79 | \( 1 + 0.231T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 1.29T + 89T^{2} \) |
| 97 | \( 1 + 9.14T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.45937506148660910348733294652, −7.12912594952681613111344498855, −6.18778433245948962747522000172, −5.53864508307510567599228003940, −4.83938728898926514023512562101, −4.32904084535442196674784784334, −3.16829135815889285820444937905, −2.35670115500420809458446424689, −1.04436527570797255526434400153, 0,
1.04436527570797255526434400153, 2.35670115500420809458446424689, 3.16829135815889285820444937905, 4.32904084535442196674784784334, 4.83938728898926514023512562101, 5.53864508307510567599228003940, 6.18778433245948962747522000172, 7.12912594952681613111344498855, 7.45937506148660910348733294652
