| L(s) = 1 | + 2.28·2-s + 3.23·4-s − 0.631·5-s + 0.582·7-s + 2.83·8-s − 1.44·10-s + 11-s − 2.96·13-s + 1.33·14-s + 0.0145·16-s − 5.10·17-s − 0.695·19-s − 2.04·20-s + 2.28·22-s − 5.73·23-s − 4.60·25-s − 6.78·26-s + 1.88·28-s − 0.254·29-s − 1.80·31-s − 5.64·32-s − 11.6·34-s − 0.368·35-s − 0.222·37-s − 1.59·38-s − 1.79·40-s + 0.991·41-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.61·4-s − 0.282·5-s + 0.220·7-s + 1.00·8-s − 0.457·10-s + 0.301·11-s − 0.822·13-s + 0.356·14-s + 0.00364·16-s − 1.23·17-s − 0.159·19-s − 0.457·20-s + 0.488·22-s − 1.19·23-s − 0.920·25-s − 1.33·26-s + 0.356·28-s − 0.0472·29-s − 0.324·31-s − 0.997·32-s − 2.00·34-s − 0.0622·35-s − 0.0365·37-s − 0.258·38-s − 0.283·40-s + 0.154·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 5 | \( 1 + 0.631T + 5T^{2} \) |
| 7 | \( 1 - 0.582T + 7T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 + 0.695T + 19T^{2} \) |
| 23 | \( 1 + 5.73T + 23T^{2} \) |
| 29 | \( 1 + 0.254T + 29T^{2} \) |
| 31 | \( 1 + 1.80T + 31T^{2} \) |
| 37 | \( 1 + 0.222T + 37T^{2} \) |
| 41 | \( 1 - 0.991T + 41T^{2} \) |
| 43 | \( 1 + 1.31T + 43T^{2} \) |
| 47 | \( 1 - 8.61T + 47T^{2} \) |
| 53 | \( 1 - 2.36T + 53T^{2} \) |
| 59 | \( 1 - 2.31T + 59T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 - 6.30T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 7.48T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 + 1.52T + 97T^{2} \) |
| 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
−7.41999481574973040876316042180, −6.85916685765729296410517252714, −6.07110972952640326256934413849, −5.53602884477664184186810245756, −4.57972401003355861614359605900, −4.24864754079442324747187726829, −3.48027707965538297304612527820, −2.48464225923944360477497648223, −1.86894527282798635143285773905, 0,
1.86894527282798635143285773905, 2.48464225923944360477497648223, 3.48027707965538297304612527820, 4.24864754079442324747187726829, 4.57972401003355861614359605900, 5.53602884477664184186810245756, 6.07110972952640326256934413849, 6.85916685765729296410517252714, 7.41999481574973040876316042180
