| L(s) = 1 | + 2-s − 4-s + 4·5-s − 2·7-s − 3·8-s + 4·10-s + 11-s − 2·13-s − 2·14-s − 16-s − 2·17-s − 6·19-s − 4·20-s + 22-s − 4·23-s + 11·25-s − 2·26-s + 2·28-s + 6·29-s + 4·31-s + 5·32-s − 2·34-s − 8·35-s − 6·37-s − 6·38-s − 12·40-s + 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.78·5-s − 0.755·7-s − 1.06·8-s + 1.26·10-s + 0.301·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.894·20-s + 0.213·22-s − 0.834·23-s + 11/5·25-s − 0.392·26-s + 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.883·32-s − 0.342·34-s − 1.35·35-s − 0.986·37-s − 0.973·38-s − 1.89·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.364292295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364292295\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.90490267946747673319167988026, −12.98872560550179677237523477012, −12.34498791478187220687210847964, −10.45031744709829536584104365809, −9.602153864880356635903308889209, −8.765403563089383948261377682178, −6.52495869918911105195591443455, −5.82642873508535905813523337142, −4.44676656528528856043784213256, −2.56141659749378916924669768053,
2.56141659749378916924669768053, 4.44676656528528856043784213256, 5.82642873508535905813523337142, 6.52495869918911105195591443455, 8.765403563089383948261377682178, 9.602153864880356635903308889209, 10.45031744709829536584104365809, 12.34498791478187220687210847964, 12.98872560550179677237523477012, 13.90490267946747673319167988026
