| L(s) = 1 | − 2·2-s − 5.77·3-s + 4·4-s + 5·5-s + 11.5·6-s − 12.7·7-s − 8·8-s + 6.31·9-s − 10·10-s + 51.7·11-s − 23.0·12-s + 33.9·13-s + 25.5·14-s − 28.8·15-s + 16·16-s − 86.4·17-s − 12.6·18-s + 59.7·19-s + 20·20-s + 73.7·21-s − 103.·22-s − 23·23-s + 46.1·24-s + 25·25-s − 67.8·26-s + 119.·27-s − 51.0·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.11·3-s + 0.5·4-s + 0.447·5-s + 0.785·6-s − 0.689·7-s − 0.353·8-s + 0.233·9-s − 0.316·10-s + 1.41·11-s − 0.555·12-s + 0.724·13-s + 0.487·14-s − 0.496·15-s + 0.250·16-s − 1.23·17-s − 0.165·18-s + 0.721·19-s + 0.223·20-s + 0.766·21-s − 1.00·22-s − 0.208·23-s + 0.392·24-s + 0.200·25-s − 0.512·26-s + 0.850·27-s − 0.344·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 5.77T + 27T^{2} \) |
| 7 | \( 1 + 12.7T + 343T^{2} \) |
| 11 | \( 1 - 51.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 86.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 64.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 275.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 482.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 270.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 320.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 627.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 34.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 90.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 182.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 73.2T + 3.89e5T^{2} \) |
| 79 | \( 1 + 283.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 476.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.82e3T + 9.12e5T^{2} \) |
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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
−11.32277614476745674035118807367, −10.33377867759958170136745116282, −9.371211236571410539328900317643, −8.581279755219757847558971592284, −6.77430176058853290361448310050, −6.44591785665238699526585567015, −5.26476880772818464526443845018, −3.54923107364091131856081569418, −1.56218787078874294915871330238, 0,
1.56218787078874294915871330238, 3.54923107364091131856081569418, 5.26476880772818464526443845018, 6.44591785665238699526585567015, 6.77430176058853290361448310050, 8.581279755219757847558971592284, 9.371211236571410539328900317643, 10.33377867759958170136745116282, 11.32277614476745674035118807367
