| L(s) = 1 | + 0.494·2-s − 3-s − 1.75·4-s − 0.494·6-s − 1.85·8-s + 9-s + 5.33·11-s + 1.75·12-s + 5.09·13-s + 2.59·16-s + 0.350·17-s + 0.494·18-s − 2.76·19-s + 2.63·22-s − 7.51·23-s + 1.85·24-s + 2.51·26-s − 27-s + 4·29-s − 6.11·31-s + 4.99·32-s − 5.33·33-s + 0.173·34-s − 1.75·36-s + 3.52·37-s − 1.36·38-s − 5.09·39-s + ⋯ |
| L(s) = 1 | + 0.349·2-s − 0.577·3-s − 0.877·4-s − 0.201·6-s − 0.656·8-s + 0.333·9-s + 1.60·11-s + 0.506·12-s + 1.41·13-s + 0.648·16-s + 0.0849·17-s + 0.116·18-s − 0.634·19-s + 0.562·22-s − 1.56·23-s + 0.378·24-s + 0.493·26-s − 0.192·27-s + 0.742·29-s − 1.09·31-s + 0.882·32-s − 0.929·33-s + 0.0296·34-s − 0.292·36-s + 0.579·37-s − 0.221·38-s − 0.815·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.542659335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542659335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.494T + 2T^{2} \) |
| 11 | \( 1 - 5.33T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 - 0.350T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 + 7.51T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 6.11T + 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 - 7.86T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 + 8.85T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 - 7.67T + 73T^{2} \) |
| 79 | \( 1 - 2.36T + 79T^{2} \) |
| 83 | \( 1 + 6.87T + 83T^{2} \) |
| 89 | \( 1 - 4.35T + 89T^{2} \) |
| 97 | \( 1 - 5.49T + 97T^{2} \) |
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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
−8.552379583717755502149981336069, −7.966142875345387627284903371194, −6.73143416609528223651335034422, −6.15428911739918075958513043457, −5.68455040462094995673312831711, −4.54389904671939813419331350747, −4.01684420068330271078517764083, −3.42658903803843866909974225979, −1.78531837338236419700659933121, −0.74819355755215240958031108992,
0.74819355755215240958031108992, 1.78531837338236419700659933121, 3.42658903803843866909974225979, 4.01684420068330271078517764083, 4.54389904671939813419331350747, 5.68455040462094995673312831711, 6.15428911739918075958513043457, 6.73143416609528223651335034422, 7.966142875345387627284903371194, 8.552379583717755502149981336069
