| L(s) = 1 | − 2.41·2-s + 3-s + 3.82·4-s − 5-s − 2.41·6-s − 0.585·7-s − 4.41·8-s + 9-s + 2.41·10-s − 2.82·11-s + 3.82·12-s − 2.58·13-s + 1.41·14-s − 15-s + 2.99·16-s − 4·17-s − 2.41·18-s + 2.82·19-s − 3.82·20-s − 0.585·21-s + 6.82·22-s − 6·23-s − 4.41·24-s + 25-s + 6.24·26-s + 27-s − 2.24·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 0.577·3-s + 1.91·4-s − 0.447·5-s − 0.985·6-s − 0.221·7-s − 1.56·8-s + 0.333·9-s + 0.763·10-s − 0.852·11-s + 1.10·12-s − 0.717·13-s + 0.377·14-s − 0.258·15-s + 0.749·16-s − 0.970·17-s − 0.569·18-s + 0.648·19-s − 0.856·20-s − 0.127·21-s + 1.45·22-s − 1.25·23-s − 0.901·24-s + 0.200·25-s + 1.22·26-s + 0.192·27-s − 0.423·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 + 0.585T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 0.242T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 3.89T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 0.343T + 83T^{2} \) |
| 89 | \( 1 - 5.07T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−10.18009326252125165046068149465, −9.741129329602803732537056492888, −8.706328387019095597966801261923, −8.043358386652882921381706620166, −7.36707277876973272180811318595, −6.42990980240923616179524075766, −4.75017997526825188335544832936, −3.10080138876143517095807165572, −1.95037852596544015350199669678, 0,
1.95037852596544015350199669678, 3.10080138876143517095807165572, 4.75017997526825188335544832936, 6.42990980240923616179524075766, 7.36707277876973272180811318595, 8.043358386652882921381706620166, 8.706328387019095597966801261923, 9.741129329602803732537056492888, 10.18009326252125165046068149465
