| L(s) = 1 | + 2·2-s + 10·3-s + 4·4-s − 5·5-s + 20·6-s + 7·7-s + 8·8-s + 73·9-s − 10·10-s − 11·11-s + 40·12-s + 68·13-s + 14·14-s − 50·15-s + 16·16-s − 72·17-s + 146·18-s + 44·19-s − 20·20-s + 70·21-s − 22·22-s + 12·23-s + 80·24-s + 25·25-s + 136·26-s + 460·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.92·3-s + 1/2·4-s − 0.447·5-s + 1.36·6-s + 0.377·7-s + 0.353·8-s + 2.70·9-s − 0.316·10-s − 0.301·11-s + 0.962·12-s + 1.45·13-s + 0.267·14-s − 0.860·15-s + 1/4·16-s − 1.02·17-s + 1.91·18-s + 0.531·19-s − 0.223·20-s + 0.727·21-s − 0.213·22-s + 0.108·23-s + 0.680·24-s + 1/5·25-s + 1.02·26-s + 3.27·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.777349609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.777349609\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 - 10 T + p^{3} T^{2} \) |
| 13 | \( 1 - 68 T + p^{3} T^{2} \) |
| 17 | \( 1 + 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 12 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 202 T + p^{3} T^{2} \) |
| 37 | \( 1 + 46 T + p^{3} T^{2} \) |
| 41 | \( 1 + 120 T + p^{3} T^{2} \) |
| 43 | \( 1 + 244 T + p^{3} T^{2} \) |
| 47 | \( 1 + 390 T + p^{3} T^{2} \) |
| 53 | \( 1 - 666 T + p^{3} T^{2} \) |
| 59 | \( 1 - 690 T + p^{3} T^{2} \) |
| 61 | \( 1 - 704 T + p^{3} T^{2} \) |
| 67 | \( 1 + 136 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1164 T + p^{3} T^{2} \) |
| 73 | \( 1 - 908 T + p^{3} T^{2} \) |
| 79 | \( 1 - 896 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1044 T + p^{3} T^{2} \) |
| 89 | \( 1 + 54 T + p^{3} T^{2} \) |
| 97 | \( 1 + 106 T + p^{3} T^{2} \) |
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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
−9.809420662079428146858261516303, −8.704895402830351269411739790534, −8.390470646887825051920209789007, −7.40991035043715271111776508746, −6.69425046147079352967652410763, −5.18231889189546758413940485158, −4.01274751348989334484337231095, −3.54684083205241536783525058329, −2.46101133055085615528096471180, −1.44632183721811002326843231477,
1.44632183721811002326843231477, 2.46101133055085615528096471180, 3.54684083205241536783525058329, 4.01274751348989334484337231095, 5.18231889189546758413940485158, 6.69425046147079352967652410763, 7.40991035043715271111776508746, 8.390470646887825051920209789007, 8.704895402830351269411739790534, 9.809420662079428146858261516303
