| L(s) = 1 | + 9·3-s − 20·5-s − 2·7-s + 54·9-s + 52·11-s + 43·13-s − 180·15-s − 50·17-s + 74·19-s − 18·21-s + 23·23-s + 275·25-s + 243·27-s − 7·29-s + 273·31-s + 468·33-s + 40·35-s − 4·37-s + 387·39-s + 123·41-s + 152·43-s − 1.08e3·45-s − 75·47-s − 339·49-s − 450·51-s + 86·53-s − 1.04e3·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.78·5-s − 0.107·7-s + 2·9-s + 1.42·11-s + 0.917·13-s − 3.09·15-s − 0.713·17-s + 0.893·19-s − 0.187·21-s + 0.208·23-s + 11/5·25-s + 1.73·27-s − 0.0448·29-s + 1.58·31-s + 2.46·33-s + 0.193·35-s − 0.0177·37-s + 1.58·39-s + 0.468·41-s + 0.539·43-s − 3.57·45-s − 0.232·47-s − 0.988·49-s − 1.23·51-s + 0.222·53-s − 2.54·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.902926577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.902926577\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - p T \) |
| good | 3 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 43 T + p^{3} T^{2} \) |
| 17 | \( 1 + 50 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 29 | \( 1 + 7 T + p^{3} T^{2} \) |
| 31 | \( 1 - 273 T + p^{3} T^{2} \) |
| 37 | \( 1 + 4 T + p^{3} T^{2} \) |
| 41 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 152 T + p^{3} T^{2} \) |
| 47 | \( 1 + 75 T + p^{3} T^{2} \) |
| 53 | \( 1 - 86 T + p^{3} T^{2} \) |
| 59 | \( 1 - 444 T + p^{3} T^{2} \) |
| 61 | \( 1 - 262 T + p^{3} T^{2} \) |
| 67 | \( 1 + 764 T + p^{3} T^{2} \) |
| 71 | \( 1 - 21 T + p^{3} T^{2} \) |
| 73 | \( 1 - 681 T + p^{3} T^{2} \) |
| 79 | \( 1 + 426 T + p^{3} T^{2} \) |
| 83 | \( 1 + 902 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1272 T + p^{3} T^{2} \) |
| 97 | \( 1 + 342 T + p^{3} T^{2} \) |
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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
−11.16446876987415938193693301323, −9.750259473808404949439997033049, −8.800186206099451382949755420525, −8.355931519801712840914130619303, −7.45401779313951707491710299058, −6.63625181624990115751908626671, −4.37752286900095525915764053717, −3.79120108632881803709822560744, −2.94631692207057357503059881801, −1.15181243010546891159543362901,
1.15181243010546891159543362901, 2.94631692207057357503059881801, 3.79120108632881803709822560744, 4.37752286900095525915764053717, 6.63625181624990115751908626671, 7.45401779313951707491710299058, 8.355931519801712840914130619303, 8.800186206099451382949755420525, 9.750259473808404949439997033049, 11.16446876987415938193693301323
