L(s) = 1 | + 3.20i·3-s + (2.02 − 0.945i)5-s − 4.54i·7-s − 7.29·9-s + 4.59·11-s − 5.17i·13-s + (3.03 + 6.50i)15-s + 3.33i·17-s + 4.43·19-s + 14.5·21-s − i·23-s + (3.21 − 3.83i)25-s − 13.8i·27-s − 4.13·29-s + 7.82·31-s + ⋯ |
L(s) = 1 | + 1.85i·3-s + (0.906 − 0.423i)5-s − 1.71i·7-s − 2.43·9-s + 1.38·11-s − 1.43i·13-s + (0.783 + 1.67i)15-s + 0.809i·17-s + 1.01·19-s + 3.18·21-s − 0.208i·23-s + (0.642 − 0.766i)25-s − 2.65i·27-s − 0.766·29-s + 1.40·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.85834 + 0.412418i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.85834 + 0.412418i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.02 + 0.945i)T \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 - 3.20iT - 3T^{2} \) |
| 7 | \( 1 + 4.54iT - 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 + 5.17iT - 13T^{2} \) |
| 17 | \( 1 - 3.33iT - 17T^{2} \) |
| 19 | \( 1 - 4.43T + 19T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 + 1.03iT - 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 43 | \( 1 + 1.67iT - 43T^{2} \) |
| 47 | \( 1 - 6.20iT - 47T^{2} \) |
| 53 | \( 1 - 7.35iT - 53T^{2} \) |
| 59 | \( 1 + 1.83T + 59T^{2} \) |
| 61 | \( 1 - 0.524T + 61T^{2} \) |
| 67 | \( 1 + 2.55iT - 67T^{2} \) |
| 71 | \( 1 + 6.58T + 71T^{2} \) |
| 73 | \( 1 - 2.41iT - 73T^{2} \) |
| 79 | \( 1 - 9.85T + 79T^{2} \) |
| 83 | \( 1 - 10.7iT - 83T^{2} \) |
| 89 | \( 1 + 5.13T + 89T^{2} \) |
| 97 | \( 1 + 3.36iT - 97T^{2} \) |
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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
−10.11518647091628847450123712402, −9.602055656104511083878853776669, −8.767095878234418256370024856479, −7.81626647513721943179902900508, −6.46318253083133177434756133368, −5.58970445199339764066942808851, −4.66784225444538387062313236432, −3.94180325591282773301594420451, −3.15431973944927297379704415503, −1.03944065497005211650234093339,
1.48702979283524138756525661691, 2.15723246134015602230845822952, 3.10759963164592093069671028013, 5.16488299348283905185178825600, 6.03344715237402053452904866791, 6.59233494257185718266107434783, 7.17513959773576315862333386596, 8.420857974426162718550185244007, 9.122006746892561409126068424387, 9.612649106585037468949258051557