| L(s) = 1 | − 3-s − 3.27·5-s + 7-s + 9-s + 1.42·11-s + 4.69·13-s + 3.27·15-s + 3.96·17-s + 5.42·19-s − 21-s − 23-s + 5.69·25-s − 27-s + 3.96·29-s + 8.81·31-s − 1.42·33-s − 3.27·35-s + 4.81·37-s − 4.69·39-s + 2.57·41-s + 11.5·43-s − 3.27·45-s − 5.39·47-s + 49-s − 3.96·51-s − 12.6·53-s − 4.66·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.46·5-s + 0.377·7-s + 0.333·9-s + 0.429·11-s + 1.30·13-s + 0.844·15-s + 0.962·17-s + 1.24·19-s − 0.218·21-s − 0.208·23-s + 1.13·25-s − 0.192·27-s + 0.736·29-s + 1.58·31-s − 0.248·33-s − 0.552·35-s + 0.792·37-s − 0.752·39-s + 0.402·41-s + 1.76·43-s − 0.487·45-s − 0.786·47-s + 0.142·49-s − 0.555·51-s − 1.73·53-s − 0.628·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.620724740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.620724740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3.27T + 5T^{2} \) |
| 11 | \( 1 - 1.42T + 11T^{2} \) |
| 13 | \( 1 - 4.69T + 13T^{2} \) |
| 17 | \( 1 - 3.96T + 17T^{2} \) |
| 19 | \( 1 - 5.42T + 19T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 - 8.81T + 31T^{2} \) |
| 37 | \( 1 - 4.81T + 37T^{2} \) |
| 41 | \( 1 - 2.57T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 5.39T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 5.51T + 67T^{2} \) |
| 71 | \( 1 - 2.69T + 71T^{2} \) |
| 73 | \( 1 + 9.35T + 73T^{2} \) |
| 79 | \( 1 - 4.57T + 79T^{2} \) |
| 83 | \( 1 - 0.277T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 7.96T + 97T^{2} \) |
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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
−7.67909149888381000644304523709, −7.49473276771038733616942292361, −6.24064871062307287237729858510, −6.03877269345023593295704421968, −4.80167731719864991315616531794, −4.43914081267001439839062008761, −3.53826642114445110750898210500, −3.01411131596091871693899868618, −1.35921471796453170117873633247, −0.75929154463493432190187908974,
0.75929154463493432190187908974, 1.35921471796453170117873633247, 3.01411131596091871693899868618, 3.53826642114445110750898210500, 4.43914081267001439839062008761, 4.80167731719864991315616531794, 6.03877269345023593295704421968, 6.24064871062307287237729858510, 7.49473276771038733616942292361, 7.67909149888381000644304523709
