| L(s) = 1 | + 0.302·3-s − 5-s + 3.30·7-s − 2.90·9-s + 1.69·11-s − 3.30·13-s − 0.302·15-s + 6.90·17-s − 5.90·19-s + 1.00·21-s − 23-s + 25-s − 1.78·27-s + 2.60·29-s − 7.90·31-s + 0.513·33-s − 3.30·35-s − 8·37-s − 1.00·39-s + 0.908·41-s + 9.21·43-s + 2.90·45-s − 2.60·47-s + 3.90·49-s + 2.09·51-s + 11.2·53-s − 1.69·55-s + ⋯ |
| L(s) = 1 | + 0.174·3-s − 0.447·5-s + 1.24·7-s − 0.969·9-s + 0.511·11-s − 0.916·13-s − 0.0781·15-s + 1.67·17-s − 1.35·19-s + 0.218·21-s − 0.208·23-s + 0.200·25-s − 0.344·27-s + 0.483·29-s − 1.42·31-s + 0.0894·33-s − 0.558·35-s − 1.31·37-s − 0.160·39-s + 0.141·41-s + 1.40·43-s + 0.433·45-s − 0.380·47-s + 0.558·49-s + 0.292·51-s + 1.53·53-s − 0.228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 0.302T + 3T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 + 5.90T + 19T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 0.908T + 41T^{2} \) |
| 43 | \( 1 - 9.21T + 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 3.39T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 6.30T + 97T^{2} \) |
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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
−7.50566752367698501639112375208, −7.18584322510690673768686935598, −5.96002464026647663803941422649, −5.48865571271591317563562964026, −4.68186054890195905046869141343, −3.99542928913818942937158401881, −3.13634902836226910176117881554, −2.24340247467640058739608131715, −1.35479781565662687111880634573, 0,
1.35479781565662687111880634573, 2.24340247467640058739608131715, 3.13634902836226910176117881554, 3.99542928913818942937158401881, 4.68186054890195905046869141343, 5.48865571271591317563562964026, 5.96002464026647663803941422649, 7.18584322510690673768686935598, 7.50566752367698501639112375208
