| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s + 7·7-s + 8·8-s + 10·10-s + 26·13-s + 14·14-s + 16·16-s − 18·17-s + 92·19-s + 20·20-s + 25·25-s + 52·26-s + 28·28-s + 6·29-s − 4·31-s + 32·32-s − 36·34-s + 35·35-s + 410·37-s + 184·38-s + 40·40-s − 174·41-s + 248·43-s − 420·47-s + 49·49-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 + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.256·17-s + 1.11·19-s + 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 0.0384·29-s − 0.0231·31-s + 0.176·32-s − 0.181·34-s + 0.169·35-s + 1.82·37-s + 0.785·38-s + 0.158·40-s − 0.662·41-s + 0.879·43-s − 1.30·47-s + 1/7·49-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
| \(L(2)\) |
\(\approx\) |
\(3.882277637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.882277637\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 4 T + p^{3} T^{2} \) |
| 37 | \( 1 - 410 T + p^{3} T^{2} \) |
| 41 | \( 1 + 174 T + p^{3} T^{2} \) |
| 43 | \( 1 - 248 T + p^{3} T^{2} \) |
| 47 | \( 1 + 420 T + p^{3} T^{2} \) |
| 53 | \( 1 + 102 T + p^{3} T^{2} \) |
| 59 | \( 1 - 588 T + p^{3} T^{2} \) |
| 61 | \( 1 - 650 T + p^{3} T^{2} \) |
| 67 | \( 1 - 152 T + p^{3} T^{2} \) |
| 71 | \( 1 - 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 610 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1048 T + p^{3} T^{2} \) |
| 83 | \( 1 - 684 T + p^{3} T^{2} \) |
| 89 | \( 1 - 834 T + p^{3} T^{2} \) |
| 97 | \( 1 - 110 T + p^{3} T^{2} \) |
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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
−10.26855065286838475663082984325, −9.427141207786832299517194046086, −8.365117314582883791493549878965, −7.43190814870170969657281818371, −6.42632788484694919866219776991, −5.58344287061491887820025729828, −4.69511910155385175312564765440, −3.58004055912206209669236922142, −2.41715403044705351124444473672, −1.13980725606096302733899326841,
1.13980725606096302733899326841, 2.41715403044705351124444473672, 3.58004055912206209669236922142, 4.69511910155385175312564765440, 5.58344287061491887820025729828, 6.42632788484694919866219776991, 7.43190814870170969657281818371, 8.365117314582883791493549878965, 9.427141207786832299517194046086, 10.26855065286838475663082984325
