| L(s) = 1 | − 3.30·3-s + 5-s − 2.94·7-s + 7.89·9-s + 0.430·11-s − 2.53·13-s − 3.30·15-s − 1.14·17-s + 1.90·19-s + 9.72·21-s − 23-s + 25-s − 16.1·27-s + 0.895·29-s + 2.05·31-s − 1.42·33-s − 2.94·35-s + 0.715·37-s + 8.35·39-s + 4.25·41-s − 7.79·43-s + 7.89·45-s + 11.7·47-s + 1.68·49-s + 3.78·51-s + 2.36·53-s + 0.430·55-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 0.447·5-s − 1.11·7-s + 2.63·9-s + 0.129·11-s − 0.702·13-s − 0.852·15-s − 0.278·17-s + 0.436·19-s + 2.12·21-s − 0.208·23-s + 0.200·25-s − 3.10·27-s + 0.166·29-s + 0.369·31-s − 0.247·33-s − 0.498·35-s + 0.117·37-s + 1.33·39-s + 0.664·41-s − 1.18·43-s + 1.17·45-s + 1.70·47-s + 0.240·49-s + 0.529·51-s + 0.325·53-s + 0.0581·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 + 2.94T + 7T^{2} \) |
| 11 | \( 1 - 0.430T + 11T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 29 | \( 1 - 0.895T + 29T^{2} \) |
| 31 | \( 1 - 2.05T + 31T^{2} \) |
| 37 | \( 1 - 0.715T + 37T^{2} \) |
| 41 | \( 1 - 4.25T + 41T^{2} \) |
| 43 | \( 1 + 7.79T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 2.36T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 3.84T + 61T^{2} \) |
| 67 | \( 1 - 4.46T + 67T^{2} \) |
| 71 | \( 1 - 0.983T + 71T^{2} \) |
| 73 | \( 1 - 8.60T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 0.182T + 83T^{2} \) |
| 89 | \( 1 - 5.54T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
| 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
−7.934939795777232604313467185996, −7.00542692897309872487017904972, −6.58594935290356741468045837093, −5.94643741859533714888116305094, −5.29170474526915566425163863607, −4.58588497603279909847973235212, −3.65159176163879450584252214618, −2.37103959468325454056500175968, −1.06162284521676019937095191271, 0,
1.06162284521676019937095191271, 2.37103959468325454056500175968, 3.65159176163879450584252214618, 4.58588497603279909847973235212, 5.29170474526915566425163863607, 5.94643741859533714888116305094, 6.58594935290356741468045837093, 7.00542692897309872487017904972, 7.934939795777232604313467185996
