| L(s) = 1 | + 2-s + 4-s + 4.16·5-s + 2.94·7-s + 8-s + 4.16·10-s − 3.28·11-s − 0.140·13-s + 2.94·14-s + 16-s + 6.25·17-s − 2.37·19-s + 4.16·20-s − 3.28·22-s + 12.3·25-s − 0.140·26-s + 2.94·28-s + 9.93·29-s + 2.28·31-s + 32-s + 6.25·34-s + 12.2·35-s − 6.20·37-s − 2.37·38-s + 4.16·40-s − 6.45·41-s + 1.28·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.86·5-s + 1.11·7-s + 0.353·8-s + 1.31·10-s − 0.990·11-s − 0.0389·13-s + 0.786·14-s + 0.250·16-s + 1.51·17-s − 0.544·19-s + 0.930·20-s − 0.700·22-s + 2.46·25-s − 0.0275·26-s + 0.556·28-s + 1.84·29-s + 0.410·31-s + 0.176·32-s + 1.07·34-s + 2.07·35-s − 1.02·37-s − 0.385·38-s + 0.657·40-s − 1.00·41-s + 0.195·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.040788710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.040788710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 4.16T + 5T^{2} \) |
| 7 | \( 1 - 2.94T + 7T^{2} \) |
| 11 | \( 1 + 3.28T + 11T^{2} \) |
| 13 | \( 1 + 0.140T + 13T^{2} \) |
| 17 | \( 1 - 6.25T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 29 | \( 1 - 9.93T + 29T^{2} \) |
| 31 | \( 1 - 2.28T + 31T^{2} \) |
| 37 | \( 1 + 6.20T + 37T^{2} \) |
| 41 | \( 1 + 6.45T + 41T^{2} \) |
| 43 | \( 1 - 1.28T + 43T^{2} \) |
| 47 | \( 1 + 2.84T + 47T^{2} \) |
| 53 | \( 1 - 4.35T + 53T^{2} \) |
| 59 | \( 1 + 0.0732T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 + 4.33T + 67T^{2} \) |
| 71 | \( 1 + 9.78T + 71T^{2} \) |
| 73 | \( 1 - 5.48T + 73T^{2} \) |
| 79 | \( 1 + 9.62T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.64742546298692944136478380196, −6.80747712970393180660368084115, −6.13751191813956376784426190931, −5.50730938904797861322235828602, −5.06048577172300566703559948047, −4.55263137174933448863491669741, −3.22873636225581868292175956828, −2.60070878169393160332688980739, −1.83148970235532111752729185053, −1.16752055111515405785216896959,
1.16752055111515405785216896959, 1.83148970235532111752729185053, 2.60070878169393160332688980739, 3.22873636225581868292175956828, 4.55263137174933448863491669741, 5.06048577172300566703559948047, 5.50730938904797861322235828602, 6.13751191813956376784426190931, 6.80747712970393180660368084115, 7.64742546298692944136478380196
