| L(s) = 1 | + 2·2-s + 4·4-s − 5·5-s + 7·7-s + 8·8-s − 10·10-s − 24·11-s + 74·13-s + 14·14-s + 16·16-s − 24·17-s − 34·19-s − 20·20-s − 48·22-s + 168·23-s + 25·25-s + 148·26-s + 28·28-s − 162·29-s + 128·31-s + 32·32-s − 48·34-s − 35·35-s + 380·37-s − 68·38-s − 40·40-s + 126·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.657·11-s + 1.57·13-s + 0.267·14-s + 1/4·16-s − 0.342·17-s − 0.410·19-s − 0.223·20-s − 0.465·22-s + 1.52·23-s + 1/5·25-s + 1.11·26-s + 0.188·28-s − 1.03·29-s + 0.741·31-s + 0.176·32-s − 0.242·34-s − 0.169·35-s + 1.68·37-s − 0.290·38-s − 0.158·40-s + 0.479·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.214944286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.214944286\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 74 T + p^{3} T^{2} \) |
| 17 | \( 1 + 24 T + p^{3} T^{2} \) |
| 19 | \( 1 + 34 T + p^{3} T^{2} \) |
| 23 | \( 1 - 168 T + p^{3} T^{2} \) |
| 29 | \( 1 + 162 T + p^{3} T^{2} \) |
| 31 | \( 1 - 128 T + p^{3} T^{2} \) |
| 37 | \( 1 - 380 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 34 T + p^{3} T^{2} \) |
| 47 | \( 1 - 294 T + p^{3} T^{2} \) |
| 53 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 444 T + p^{3} T^{2} \) |
| 61 | \( 1 + 592 T + p^{3} T^{2} \) |
| 67 | \( 1 - 110 T + p^{3} T^{2} \) |
| 71 | \( 1 + 198 T + p^{3} T^{2} \) |
| 73 | \( 1 - 866 T + p^{3} T^{2} \) |
| 79 | \( 1 - 776 T + p^{3} T^{2} \) |
| 83 | \( 1 + 576 T + p^{3} T^{2} \) |
| 89 | \( 1 + 354 T + p^{3} T^{2} \) |
| 97 | \( 1 - 614 T + p^{3} T^{2} \) |
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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
−10.63334710544528190904571508624, −9.209965228528570860588495656627, −8.334048990653452724558547040246, −7.51161418211733936666610915767, −6.48228815628587559598274266475, −5.57644805811745539166983356497, −4.55087851364028886375480086287, −3.66205104917157337219118779226, −2.50816510259442547823352222052, −0.999776264958444754805440245108,
0.999776264958444754805440245108, 2.50816510259442547823352222052, 3.66205104917157337219118779226, 4.55087851364028886375480086287, 5.57644805811745539166983356497, 6.48228815628587559598274266475, 7.51161418211733936666610915767, 8.334048990653452724558547040246, 9.209965228528570860588495656627, 10.63334710544528190904571508624
