| L(s) = 1 | − 2.34·2-s + 1.80·3-s + 3.52·4-s − 4.24·6-s + 4.19·7-s − 3.57·8-s + 0.263·9-s − 2.93·11-s + 6.35·12-s − 1.23·13-s − 9.86·14-s + 1.35·16-s + 4.88·17-s − 0.619·18-s − 4.31·19-s + 7.58·21-s + 6.89·22-s + 1.20·23-s − 6.45·24-s + 2.89·26-s − 4.94·27-s + 14.7·28-s + 4.59·29-s − 2.00·31-s + 3.96·32-s − 5.29·33-s − 11.4·34-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.04·3-s + 1.76·4-s − 1.73·6-s + 1.58·7-s − 1.26·8-s + 0.0878·9-s − 0.884·11-s + 1.83·12-s − 0.341·13-s − 2.63·14-s + 0.337·16-s + 1.18·17-s − 0.145·18-s − 0.990·19-s + 1.65·21-s + 1.46·22-s + 0.250·23-s − 1.31·24-s + 0.567·26-s − 0.951·27-s + 2.79·28-s + 0.852·29-s − 0.360·31-s + 0.701·32-s − 0.922·33-s − 1.96·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.413690549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.413690549\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 3 | \( 1 - 1.80T + 3T^{2} \) |
| 7 | \( 1 - 4.19T + 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 4.88T + 17T^{2} \) |
| 19 | \( 1 + 4.31T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 - 4.59T + 29T^{2} \) |
| 31 | \( 1 + 2.00T + 31T^{2} \) |
| 37 | \( 1 + 3.72T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 - 5.82T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 3.47T + 53T^{2} \) |
| 59 | \( 1 + 2.88T + 59T^{2} \) |
| 61 | \( 1 + 5.40T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 6.50T + 71T^{2} \) |
| 73 | \( 1 - 8.66T + 73T^{2} \) |
| 79 | \( 1 + 0.141T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 0.378T + 97T^{2} \) |
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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
−8.140390520150351736427122056645, −7.68447599164668997322079502747, −7.34718139602564689445078732430, −6.15326590898628089673426893544, −5.22300339900757146929559902599, −4.44334928525471954227239076311, −3.23347153390004938023677129772, −2.33538947723241709668203270038, −1.86503355721047761985493210463, −0.76526092512813862608375468460,
0.76526092512813862608375468460, 1.86503355721047761985493210463, 2.33538947723241709668203270038, 3.23347153390004938023677129772, 4.44334928525471954227239076311, 5.22300339900757146929559902599, 6.15326590898628089673426893544, 7.34718139602564689445078732430, 7.68447599164668997322079502747, 8.140390520150351736427122056645
