| L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s + 2·13-s − 4·14-s + 16-s − 6·17-s − 4·19-s + 20-s + 25-s + 2·26-s − 4·28-s + 6·29-s + 8·31-s + 32-s − 6·34-s − 4·35-s + 2·37-s − 4·38-s + 40-s + 6·41-s − 4·43-s + 9·49-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.676·35-s + 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + 9/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.337595994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.337595994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.78806788970128105043488414911, −13.19720570989419308005927819605, −12.30986793353697911256504493943, −10.91913531404329383721387224500, −9.880541566397139490214892173475, −8.644229019078172602684577487278, −6.73106470746564203527567325446, −6.11397905422548756742635703073, −4.32687879913691587957737883725, −2.74628747875934588873509462774,
2.74628747875934588873509462774, 4.32687879913691587957737883725, 6.11397905422548756742635703073, 6.73106470746564203527567325446, 8.644229019078172602684577487278, 9.880541566397139490214892173475, 10.91913531404329383721387224500, 12.30986793353697911256504493943, 13.19720570989419308005927819605, 13.78806788970128105043488414911
