| L(s) = 1 | − 2.30·3-s + 0.270·5-s + 2.30·9-s + 2.35·11-s − 5.19·13-s − 0.623·15-s + 1.62·17-s − 19-s − 3.19·23-s − 4.92·25-s + 1.60·27-s + 6.45·29-s + 2.77·31-s − 5.41·33-s + 4.97·37-s + 11.9·39-s + 4.45·41-s + 8.01·43-s + 0.623·45-s − 6.46·47-s − 3.73·51-s − 6.84·53-s + 0.637·55-s + 2.30·57-s − 2.37·59-s − 9.61·61-s − 1.40·65-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 0.121·5-s + 0.767·9-s + 0.709·11-s − 1.44·13-s − 0.161·15-s + 0.393·17-s − 0.229·19-s − 0.666·23-s − 0.985·25-s + 0.308·27-s + 1.19·29-s + 0.497·31-s − 0.943·33-s + 0.818·37-s + 1.91·39-s + 0.694·41-s + 1.22·43-s + 0.0929·45-s − 0.943·47-s − 0.523·51-s − 0.940·53-s + 0.0859·55-s + 0.305·57-s − 0.308·59-s − 1.23·61-s − 0.174·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 - 0.270T + 5T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 13 | \( 1 + 5.19T + 13T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 23 | \( 1 + 3.19T + 23T^{2} \) |
| 29 | \( 1 - 6.45T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 - 4.97T + 37T^{2} \) |
| 41 | \( 1 - 4.45T + 41T^{2} \) |
| 43 | \( 1 - 8.01T + 43T^{2} \) |
| 47 | \( 1 + 6.46T + 47T^{2} \) |
| 53 | \( 1 + 6.84T + 53T^{2} \) |
| 59 | \( 1 + 2.37T + 59T^{2} \) |
| 61 | \( 1 + 9.61T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 5.67T + 71T^{2} \) |
| 73 | \( 1 - 0.490T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 3.09T + 83T^{2} \) |
| 89 | \( 1 - 3.81T + 89T^{2} \) |
| 97 | \( 1 - 5.89T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55904168779736402361704588275, −6.54946492898954381314137859492, −6.21424064679114366285540543369, −5.53401668205612098669793354485, −4.68460997912558125480445602045, −4.32393752514475872502790856763, −3.10507654930914653446044853821, −2.15483420708713227562852651062, −1.03762168456922806237921712067, 0,
1.03762168456922806237921712067, 2.15483420708713227562852651062, 3.10507654930914653446044853821, 4.32393752514475872502790856763, 4.68460997912558125480445602045, 5.53401668205612098669793354485, 6.21424064679114366285540543369, 6.54946492898954381314137859492, 7.55904168779736402361704588275
