| L(s) = 1 | + 2.47·2-s − 3-s + 4.11·4-s − 4.22·5-s − 2.47·6-s + 1.64·7-s + 5.22·8-s + 9-s − 10.4·10-s − 2.35·11-s − 4.11·12-s − 2.58·13-s + 4.06·14-s + 4.22·15-s + 4.70·16-s + 5.87·17-s + 2.47·18-s + 2.94·19-s − 17.4·20-s − 1.64·21-s − 5.83·22-s − 2.22·23-s − 5.22·24-s + 12.8·25-s − 6.39·26-s − 27-s + 6.75·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.577·3-s + 2.05·4-s − 1.89·5-s − 1.00·6-s + 0.620·7-s + 1.84·8-s + 0.333·9-s − 3.30·10-s − 0.710·11-s − 1.18·12-s − 0.717·13-s + 1.08·14-s + 1.09·15-s + 1.17·16-s + 1.42·17-s + 0.582·18-s + 0.675·19-s − 3.89·20-s − 0.358·21-s − 1.24·22-s − 0.464·23-s − 1.06·24-s + 2.57·25-s − 1.25·26-s − 0.192·27-s + 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.648193356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.648193356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 5 | \( 1 + 4.22T + 5T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 + 0.945T + 37T^{2} \) |
| 41 | \( 1 - 0.568T + 41T^{2} \) |
| 43 | \( 1 + 2.56T + 43T^{2} \) |
| 47 | \( 1 + 4.92T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 + 7.28T + 59T^{2} \) |
| 61 | \( 1 - 4.94T + 61T^{2} \) |
| 67 | \( 1 - 0.926T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 0.945T + 73T^{2} \) |
| 79 | \( 1 + 5.51T + 79T^{2} \) |
| 83 | \( 1 - 5.17T + 83T^{2} \) |
| 89 | \( 1 - 8.47T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−14.37238333998942639646352004116, −12.91716162531684899506489271394, −11.93819997665332638897588759917, −11.70003529826021811099898227533, −10.51410641162841244031060632683, −7.922820677970911516553643364350, −7.18437778548053477104543419734, −5.42430910976235794358093330333, −4.52961583187061814094193334521, −3.30630143514452427642507298744,
3.30630143514452427642507298744, 4.52961583187061814094193334521, 5.42430910976235794358093330333, 7.18437778548053477104543419734, 7.922820677970911516553643364350, 10.51410641162841244031060632683, 11.70003529826021811099898227533, 11.93819997665332638897588759917, 12.91716162531684899506489271394, 14.37238333998942639646352004116
