| L(s) = 1 | + 4·2-s + 16·4-s − 9·5-s + 49·7-s + 64·8-s − 36·10-s − 153·11-s − 550·13-s + 196·14-s + 256·16-s + 138·17-s + 2.26e3·19-s − 144·20-s − 612·22-s + 4.77e3·23-s − 3.04e3·25-s − 2.20e3·26-s + 784·28-s + 3.31e3·29-s + 9.21e3·31-s + 1.02e3·32-s + 552·34-s − 441·35-s − 1.42e4·37-s + 9.04e3·38-s − 576·40-s − 1.66e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.160·5-s + 0.377·7-s + 0.353·8-s − 0.113·10-s − 0.381·11-s − 0.902·13-s + 0.267·14-s + 1/4·16-s + 0.115·17-s + 1.43·19-s − 0.0804·20-s − 0.269·22-s + 1.88·23-s − 0.974·25-s − 0.638·26-s + 0.188·28-s + 0.732·29-s + 1.72·31-s + 0.176·32-s + 0.0818·34-s − 0.0608·35-s − 1.71·37-s + 1.01·38-s − 0.0569·40-s − 1.54·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.468539948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.468539948\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 9 T + p^{5} T^{2} \) |
| 11 | \( 1 + 153 T + p^{5} T^{2} \) |
| 13 | \( 1 + 550 T + p^{5} T^{2} \) |
| 17 | \( 1 - 138 T + p^{5} T^{2} \) |
| 19 | \( 1 - 119 p T + p^{5} T^{2} \) |
| 23 | \( 1 - 4779 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3318 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9215 T + p^{5} T^{2} \) |
| 37 | \( 1 + 385 p T + p^{5} T^{2} \) |
| 41 | \( 1 + 16665 T + p^{5} T^{2} \) |
| 43 | \( 1 - 12884 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1284 T + p^{5} T^{2} \) |
| 53 | \( 1 - 34752 T + p^{5} T^{2} \) |
| 59 | \( 1 - 41928 T + p^{5} T^{2} \) |
| 61 | \( 1 - 22712 T + p^{5} T^{2} \) |
| 67 | \( 1 + 12034 T + p^{5} T^{2} \) |
| 71 | \( 1 + 58977 T + p^{5} T^{2} \) |
| 73 | \( 1 - 32960 T + p^{5} T^{2} \) |
| 79 | \( 1 - 75812 T + p^{5} T^{2} \) |
| 83 | \( 1 - 85974 T + p^{5} T^{2} \) |
| 89 | \( 1 - 136167 T + p^{5} T^{2} \) |
| 97 | \( 1 + 86008 T + p^{5} T^{2} \) |
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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
−10.61264229513418725352639152599, −9.817468070258341878185054191896, −8.580448786734397053937843075772, −7.53085008003606113497223203863, −6.80690062137212924554843257583, −5.37224410665828179427677567370, −4.83962776719700918897080229485, −3.46343988628237938327603520239, −2.43388709880974196127407739489, −0.922407343486427553080830031058,
0.922407343486427553080830031058, 2.43388709880974196127407739489, 3.46343988628237938327603520239, 4.83962776719700918897080229485, 5.37224410665828179427677567370, 6.80690062137212924554843257583, 7.53085008003606113497223203863, 8.580448786734397053937843075772, 9.817468070258341878185054191896, 10.61264229513418725352639152599
