| L(s) = 1 | + 2.27·2-s + 1.88·3-s + 3.16·4-s + 1.72·5-s + 4.29·6-s + 2.65·8-s + 0.569·9-s + 3.91·10-s − 2.87·11-s + 5.98·12-s − 1.56·13-s + 3.25·15-s − 0.308·16-s + 5.43·17-s + 1.29·18-s + 19-s + 5.45·20-s − 6.52·22-s − 6.09·23-s + 5.00·24-s − 2.03·25-s − 3.55·26-s − 4.59·27-s + 3.17·29-s + 7.39·30-s − 1.03·31-s − 6.00·32-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 1.09·3-s + 1.58·4-s + 0.770·5-s + 1.75·6-s + 0.937·8-s + 0.189·9-s + 1.23·10-s − 0.865·11-s + 1.72·12-s − 0.434·13-s + 0.840·15-s − 0.0770·16-s + 1.31·17-s + 0.304·18-s + 0.229·19-s + 1.21·20-s − 1.39·22-s − 1.27·23-s + 1.02·24-s − 0.406·25-s − 0.697·26-s − 0.883·27-s + 0.589·29-s + 1.35·30-s − 0.185·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.471995659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.471995659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 3 | \( 1 - 1.88T + 3T^{2} \) |
| 5 | \( 1 - 1.72T + 5T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 + 1.56T + 13T^{2} \) |
| 17 | \( 1 - 5.43T + 17T^{2} \) |
| 23 | \( 1 + 6.09T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 + 1.03T + 31T^{2} \) |
| 37 | \( 1 - 8.44T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 3.21T + 43T^{2} \) |
| 47 | \( 1 + 5.50T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 6.21T + 59T^{2} \) |
| 61 | \( 1 - 9.76T + 61T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 5.38T + 83T^{2} \) |
| 89 | \( 1 - 7.45T + 89T^{2} \) |
| 97 | \( 1 + 2.45T + 97T^{2} \) |
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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
−9.925013592126959802653449747815, −9.450356920981666627323636785680, −8.093888587974851617161432977805, −7.59812509508328156192751943663, −6.22014795811398566966720924020, −5.65130546797318211428957060626, −4.73285620902844094697603089551, −3.61720833274936652968965475696, −2.80694863951078837988347322054, −2.06004345239289758220673116380,
2.06004345239289758220673116380, 2.80694863951078837988347322054, 3.61720833274936652968965475696, 4.73285620902844094697603089551, 5.65130546797318211428957060626, 6.22014795811398566966720924020, 7.59812509508328156192751943663, 8.093888587974851617161432977805, 9.450356920981666627323636785680, 9.925013592126959802653449747815
