| L(s) = 1 | − 2-s − 31·4-s + 92·5-s − 26·7-s + 63·8-s − 92·10-s − 121·11-s − 692·13-s + 26·14-s + 929·16-s + 1.44e3·17-s + 2.16e3·19-s − 2.85e3·20-s + 121·22-s + 1.58e3·23-s + 5.33e3·25-s + 692·26-s + 806·28-s + 5.52e3·29-s + 4.79e3·31-s − 2.94e3·32-s − 1.44e3·34-s − 2.39e3·35-s − 1.01e4·37-s − 2.16e3·38-s + 5.79e3·40-s + 1.06e4·41-s + ⋯ |
| L(s) = 1 | − 0.176·2-s − 0.968·4-s + 1.64·5-s − 0.200·7-s + 0.348·8-s − 0.290·10-s − 0.301·11-s − 1.13·13-s + 0.0354·14-s + 0.907·16-s + 1.21·17-s + 1.37·19-s − 1.59·20-s + 0.0533·22-s + 0.623·23-s + 1.70·25-s + 0.200·26-s + 0.194·28-s + 1.22·29-s + 0.895·31-s − 0.508·32-s − 0.213·34-s − 0.330·35-s − 1.22·37-s − 0.242·38-s + 0.572·40-s + 0.986·41-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
| \(L(3)\) |
\(\approx\) |
\(1.76514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76514\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + T + p^{5} T^{2} \) |
| 5 | \( 1 - 92 T + p^{5} T^{2} \) |
| 7 | \( 1 + 26 T + p^{5} T^{2} \) |
| 13 | \( 1 + 692 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1442 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2160 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1582 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5526 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4792 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10194 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10622 T + p^{5} T^{2} \) |
| 43 | \( 1 - 8580 T + p^{5} T^{2} \) |
| 47 | \( 1 - 2362 T + p^{5} T^{2} \) |
| 53 | \( 1 - 30804 T + p^{5} T^{2} \) |
| 59 | \( 1 + 6416 T + p^{5} T^{2} \) |
| 61 | \( 1 - 42096 T + p^{5} T^{2} \) |
| 67 | \( 1 + 28444 T + p^{5} T^{2} \) |
| 71 | \( 1 + 45690 T + p^{5} T^{2} \) |
| 73 | \( 1 + 18374 T + p^{5} T^{2} \) |
| 79 | \( 1 + 105214 T + p^{5} T^{2} \) |
| 83 | \( 1 + 62292 T + p^{5} T^{2} \) |
| 89 | \( 1 - 72246 T + p^{5} T^{2} \) |
| 97 | \( 1 - 79262 T + p^{5} T^{2} \) |
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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
−13.12084720029171735598519572642, −12.14443880496896017704111034409, −10.13695892938027687712688929295, −9.897198760689012823339660857853, −8.826671232216666805418971513375, −7.36084609443185827375893685154, −5.73677102048045892481498855565, −4.93640382348827436648231630641, −2.86622213204796619008894915820, −1.06700858021226337619949574352,
1.06700858021226337619949574352, 2.86622213204796619008894915820, 4.93640382348827436648231630641, 5.73677102048045892481498855565, 7.36084609443185827375893685154, 8.826671232216666805418971513375, 9.897198760689012823339660857853, 10.13695892938027687712688929295, 12.14443880496896017704111034409, 13.12084720029171735598519572642
