| L(s) = 1 | + 2·2-s − 28·4-s − 46·5-s + 148·7-s − 120·8-s − 92·10-s − 121·11-s + 574·13-s + 296·14-s + 656·16-s + 722·17-s + 2.16e3·19-s + 1.28e3·20-s − 242·22-s + 2.53e3·23-s − 1.00e3·25-s + 1.14e3·26-s − 4.14e3·28-s − 4.65e3·29-s + 5.03e3·31-s + 5.15e3·32-s + 1.44e3·34-s − 6.80e3·35-s + 8.11e3·37-s + 4.32e3·38-s + 5.52e3·40-s + 5.13e3·41-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s − 0.822·5-s + 1.14·7-s − 0.662·8-s − 0.290·10-s − 0.301·11-s + 0.942·13-s + 0.403·14-s + 0.640·16-s + 0.605·17-s + 1.37·19-s + 0.720·20-s − 0.106·22-s + 0.999·23-s − 0.322·25-s + 0.333·26-s − 0.998·28-s − 1.02·29-s + 0.940·31-s + 0.889·32-s + 0.214·34-s − 0.939·35-s + 0.974·37-s + 0.485·38-s + 0.545·40-s + 0.477·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.719792622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719792622\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - p T + p^{5} T^{2} \) |
| 5 | \( 1 + 46 T + p^{5} T^{2} \) |
| 7 | \( 1 - 148 T + p^{5} T^{2} \) |
| 13 | \( 1 - 574 T + p^{5} T^{2} \) |
| 17 | \( 1 - 722 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2160 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2536 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4650 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5032 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8118 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5138 T + p^{5} T^{2} \) |
| 43 | \( 1 - 8304 T + p^{5} T^{2} \) |
| 47 | \( 1 + 24728 T + p^{5} T^{2} \) |
| 53 | \( 1 - 28746 T + p^{5} T^{2} \) |
| 59 | \( 1 - 5860 T + p^{5} T^{2} \) |
| 61 | \( 1 + 53658 T + p^{5} T^{2} \) |
| 67 | \( 1 - 30908 T + p^{5} T^{2} \) |
| 71 | \( 1 - 69648 T + p^{5} T^{2} \) |
| 73 | \( 1 + 18446 T + p^{5} T^{2} \) |
| 79 | \( 1 + 25300 T + p^{5} T^{2} \) |
| 83 | \( 1 - 17556 T + p^{5} T^{2} \) |
| 89 | \( 1 + 132570 T + p^{5} T^{2} \) |
| 97 | \( 1 - 70658 T + p^{5} T^{2} \) |
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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
−13.05734301128714713691840642515, −11.84314395745443988613761054864, −11.09538518759803285615140651951, −9.582071304332225705973765543564, −8.363785537409016563560906226795, −7.61836708007161810561839375668, −5.64031071606601308664760691384, −4.58524549375312206655816330860, −3.39812675332577800811494953680, −0.964007254837947617902024564195,
0.964007254837947617902024564195, 3.39812675332577800811494953680, 4.58524549375312206655816330860, 5.64031071606601308664760691384, 7.61836708007161810561839375668, 8.363785537409016563560906226795, 9.582071304332225705973765543564, 11.09538518759803285615140651951, 11.84314395745443988613761054864, 13.05734301128714713691840642515
