| L(s) = 1 | − 0.865i·5-s + 4.63i·7-s − 5.14i·11-s + (−1.47 + 3.28i)13-s − 3.67·17-s + 5.37i·19-s + 1.18·23-s + 4.25·25-s − 0.638·29-s − 7.68i·31-s + 4.01·35-s + 0.0776i·37-s + 8.50i·41-s − 11.7·43-s + 1.32i·47-s + ⋯ |
| L(s) = 1 | − 0.386i·5-s + 1.75i·7-s − 1.55i·11-s + (−0.409 + 0.912i)13-s − 0.891·17-s + 1.23i·19-s + 0.247·23-s + 0.850·25-s − 0.118·29-s − 1.38i·31-s + 0.678·35-s + 0.0127i·37-s + 1.32i·41-s − 1.78·43-s + 0.193i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6393285712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6393285712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (1.47 - 3.28i)T \) |
| good | 5 | \( 1 + 0.865iT - 5T^{2} \) |
| 7 | \( 1 - 4.63iT - 7T^{2} \) |
| 11 | \( 1 + 5.14iT - 11T^{2} \) |
| 17 | \( 1 + 3.67T + 17T^{2} \) |
| 19 | \( 1 - 5.37iT - 19T^{2} \) |
| 23 | \( 1 - 1.18T + 23T^{2} \) |
| 29 | \( 1 + 0.638T + 29T^{2} \) |
| 31 | \( 1 + 7.68iT - 31T^{2} \) |
| 37 | \( 1 - 0.0776iT - 37T^{2} \) |
| 41 | \( 1 - 8.50iT - 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 1.32iT - 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 5.05iT - 59T^{2} \) |
| 61 | \( 1 + 8.82T + 61T^{2} \) |
| 67 | \( 1 - 2.15iT - 67T^{2} \) |
| 71 | \( 1 + 11.5iT - 71T^{2} \) |
| 73 | \( 1 - 1.81iT - 73T^{2} \) |
| 79 | \( 1 - 8.66T + 79T^{2} \) |
| 83 | \( 1 + 4.18iT - 83T^{2} \) |
| 89 | \( 1 - 17.4iT - 89T^{2} \) |
| 97 | \( 1 + 4.99iT - 97T^{2} \) |
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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
−9.180691658526576802741539012123, −8.379948744109738989858478549692, −7.967696703928165492084687632205, −6.51608037570521279725578034950, −6.12904899102714973937001588719, −5.31779481995526162704502307816, −4.56867669206255721510902122353, −3.38896822851814704503456660686, −2.54818273780248393140425162230, −1.55881758100338698449140822274,
0.19719027408498603071391569802, 1.54693110449149221069550276409, 2.76024405191719870392788091087, 3.66494415648181306636165683061, 4.72347865346255575540657468434, 4.95548756198119961304679491591, 6.60922263774153459233331359898, 7.00253750725407357850000058869, 7.43578883913225563674591565514, 8.384780505380400826487128059621
