| L(s) = 1 | − 1.96·2-s − 3.23·3-s + 1.85·4-s + 6.34·6-s + 1.21·7-s + 0.290·8-s + 7.46·9-s + 3.74·11-s − 5.99·12-s + 3.08·13-s − 2.38·14-s − 4.27·16-s − 7.01·17-s − 14.6·18-s − 2.30·19-s − 3.92·21-s − 7.35·22-s − 1.94·23-s − 0.941·24-s − 6.05·26-s − 14.4·27-s + 2.24·28-s + 5.85·29-s − 4.10·31-s + 7.80·32-s − 12.1·33-s + 13.7·34-s + ⋯ |
| L(s) = 1 | − 1.38·2-s − 1.86·3-s + 0.925·4-s + 2.59·6-s + 0.458·7-s + 0.102·8-s + 2.48·9-s + 1.12·11-s − 1.72·12-s + 0.854·13-s − 0.636·14-s − 1.06·16-s − 1.70·17-s − 3.45·18-s − 0.528·19-s − 0.857·21-s − 1.56·22-s − 0.405·23-s − 0.192·24-s − 1.18·26-s − 2.78·27-s + 0.424·28-s + 1.08·29-s − 0.736·31-s + 1.38·32-s − 2.10·33-s + 2.36·34-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
| \(L(1)\) |
\(\approx\) |
\(0.4738078035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4738078035\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.96T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 7 | \( 1 - 1.21T + 7T^{2} \) |
| 11 | \( 1 - 3.74T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 + 7.01T + 17T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 + 1.94T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 + 4.10T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 - 2.18T + 43T^{2} \) |
| 47 | \( 1 - 9.94T + 47T^{2} \) |
| 53 | \( 1 - 0.642T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 7.63T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 3.64T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 4.41T + 79T^{2} \) |
| 83 | \( 1 + 8.23T + 83T^{2} \) |
| 89 | \( 1 - 6.52T + 89T^{2} \) |
| 97 | \( 1 + 1.16T + 97T^{2} \) |
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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
−8.189297002775397243758566145920, −7.25506831325544748647603135656, −6.55719088491634325095039688508, −6.34940308980884072112030392751, −5.36772305097391318047391843305, −4.34000736116332502078429102909, −4.16480629486349754430305687127, −2.14392771378287677076908746116, −1.31737599258431705416784608573, −0.56670075767768302415780129074,
0.56670075767768302415780129074, 1.31737599258431705416784608573, 2.14392771378287677076908746116, 4.16480629486349754430305687127, 4.34000736116332502078429102909, 5.36772305097391318047391843305, 6.34940308980884072112030392751, 6.55719088491634325095039688508, 7.25506831325544748647603135656, 8.189297002775397243758566145920
