| L(s) = 1 | + 2-s + 4-s + 5-s − 1.45·7-s + 8-s + 10-s + 3.50·11-s + 4.95·13-s − 1.45·14-s + 16-s + 3.19·17-s + 3.19·19-s + 20-s + 3.50·22-s − 4.15·23-s + 25-s + 4.95·26-s − 1.45·28-s + 3.25·29-s + 6.95·31-s + 32-s + 3.19·34-s − 1.45·35-s − 4.70·37-s + 3.19·38-s + 40-s − 8.21·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.548·7-s + 0.353·8-s + 0.316·10-s + 1.05·11-s + 1.37·13-s − 0.387·14-s + 0.250·16-s + 0.774·17-s + 0.732·19-s + 0.223·20-s + 0.748·22-s − 0.866·23-s + 0.200·25-s + 0.972·26-s − 0.274·28-s + 0.603·29-s + 1.25·31-s + 0.176·32-s + 0.547·34-s − 0.245·35-s − 0.773·37-s + 0.518·38-s + 0.158·40-s − 1.28·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.088038693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.088038693\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 - 3.50T + 11T^{2} \) |
| 13 | \( 1 - 4.95T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 + 4.15T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 - 6.95T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 + 8.21T + 41T^{2} \) |
| 43 | \( 1 + 7.01T + 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 + 0.900T + 53T^{2} \) |
| 59 | \( 1 + 8.21T + 59T^{2} \) |
| 61 | \( 1 - 0.549T + 61T^{2} \) |
| 71 | \( 1 - 6.64T + 71T^{2} \) |
| 73 | \( 1 - 7.40T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 3.50T + 97T^{2} \) |
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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
−8.137607648416466910642709241274, −7.09446130712988354304759811238, −6.43684758541010429171879363814, −6.05310109088027392232128140833, −5.27855666441404565704171590217, −4.38168343531070270312768135012, −3.52598702621087233571197751013, −3.12906884707229505517340483580, −1.83442378977609818111514535246, −1.03682371434504502760165639563,
1.03682371434504502760165639563, 1.83442378977609818111514535246, 3.12906884707229505517340483580, 3.52598702621087233571197751013, 4.38168343531070270312768135012, 5.27855666441404565704171590217, 6.05310109088027392232128140833, 6.43684758541010429171879363814, 7.09446130712988354304759811238, 8.137607648416466910642709241274
