| L(s) = 1 | + 0.473i·2-s + 1.77·4-s + 2.56i·7-s + 1.78i·8-s − 6.16·11-s − 2.13i·13-s − 1.21·14-s + 2.70·16-s + 3.16i·17-s − 0.356·19-s − 2.91i·22-s + 4.21i·23-s + 1.00·26-s + 4.55i·28-s − 1.68·29-s + ⋯ |
| L(s) = 1 | + 0.334i·2-s + 0.888·4-s + 0.968i·7-s + 0.631i·8-s − 1.85·11-s − 0.591i·13-s − 0.324·14-s + 0.676·16-s + 0.768i·17-s − 0.0817·19-s − 0.622i·22-s + 0.878i·23-s + 0.197·26-s + 0.860i·28-s − 0.313·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.214871502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214871502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.473iT - 2T^{2} \) |
| 7 | \( 1 - 2.56iT - 7T^{2} \) |
| 11 | \( 1 + 6.16T + 11T^{2} \) |
| 13 | \( 1 + 2.13iT - 13T^{2} \) |
| 17 | \( 1 - 3.16iT - 17T^{2} \) |
| 19 | \( 1 + 0.356T + 19T^{2} \) |
| 23 | \( 1 - 4.21iT - 23T^{2} \) |
| 29 | \( 1 + 1.68T + 29T^{2} \) |
| 31 | \( 1 + 8.25T + 31T^{2} \) |
| 37 | \( 1 - 3.63iT - 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 - 7.67iT - 43T^{2} \) |
| 47 | \( 1 + 11.4iT - 47T^{2} \) |
| 53 | \( 1 - 9.43iT - 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 0.0109T + 61T^{2} \) |
| 67 | \( 1 - 0.982iT - 67T^{2} \) |
| 71 | \( 1 + 6.43T + 71T^{2} \) |
| 73 | \( 1 + 6.61iT - 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 - 10.4iT - 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 - 7.20iT - 97T^{2} \) |
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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
−9.422336068137157663075115931790, −8.451849935161767410214418666527, −7.85568911291664406372467668058, −7.29834817684744515787260088608, −6.15432827060297137094512554172, −5.58361964534460246762532512207, −5.03594626113264859829864309036, −3.42402840280227718299451747433, −2.63679437891508466686820257125, −1.79416562186639510609692881839,
0.37548616613330096952572053691, 1.88614411546273063315691791618, 2.74082218441246664874929194333, 3.67307300956387166792815452353, 4.72509282502336001191454746792, 5.60349832486473857510208238489, 6.59441024224015423301369040956, 7.43135375208477146584867389907, 7.67401442976727249469760940112, 8.855635671052905757821043207558
