| L(s) = 1 | + 0.766·2-s + 3-s − 1.41·4-s − 1.12·5-s + 0.766·6-s − 2.61·8-s + 9-s − 0.864·10-s + 0.505·11-s − 1.41·12-s + 6.75·13-s − 1.12·15-s + 0.821·16-s − 3.67·17-s + 0.766·18-s − 5.76·19-s + 1.59·20-s + 0.387·22-s − 4.22·23-s − 2.61·24-s − 3.72·25-s + 5.17·26-s + 27-s + 6.10·29-s − 0.864·30-s + 6.07·31-s + 5.85·32-s + ⋯ |
| L(s) = 1 | + 0.541·2-s + 0.577·3-s − 0.706·4-s − 0.504·5-s + 0.312·6-s − 0.924·8-s + 0.333·9-s − 0.273·10-s + 0.152·11-s − 0.407·12-s + 1.87·13-s − 0.291·15-s + 0.205·16-s − 0.890·17-s + 0.180·18-s − 1.32·19-s + 0.356·20-s + 0.0826·22-s − 0.881·23-s − 0.533·24-s − 0.745·25-s + 1.01·26-s + 0.192·27-s + 1.13·29-s − 0.157·30-s + 1.09·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 0.766T + 2T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 11 | \( 1 - 0.505T + 11T^{2} \) |
| 13 | \( 1 - 6.75T + 13T^{2} \) |
| 17 | \( 1 + 3.67T + 17T^{2} \) |
| 19 | \( 1 + 5.76T + 19T^{2} \) |
| 23 | \( 1 + 4.22T + 23T^{2} \) |
| 29 | \( 1 - 6.10T + 29T^{2} \) |
| 31 | \( 1 - 6.07T + 31T^{2} \) |
| 37 | \( 1 + 3.42T + 37T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 - 3.36T + 53T^{2} \) |
| 59 | \( 1 - 8.83T + 59T^{2} \) |
| 61 | \( 1 - 9.19T + 61T^{2} \) |
| 67 | \( 1 + 4.42T + 67T^{2} \) |
| 71 | \( 1 + 7.27T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 - 9.39T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
| 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
−8.145341961475725990123672358203, −6.77542725038049031766844314982, −6.34884524987490329547297693991, −5.55459447305317850225675664453, −4.45310000268783959075647064336, −4.09402179684484561070040852954, −3.50489531105935926443402125501, −2.54155001581693189373179491158, −1.37043932474548682311062930715, 0,
1.37043932474548682311062930715, 2.54155001581693189373179491158, 3.50489531105935926443402125501, 4.09402179684484561070040852954, 4.45310000268783959075647064336, 5.55459447305317850225675664453, 6.34884524987490329547297693991, 6.77542725038049031766844314982, 8.145341961475725990123672358203
