| L(s) = 1 | + 3.37i·3-s − 2.52i·5-s + 3.31·7-s − 8.37·9-s − 2.37i·11-s − 5.84i·13-s + 8.51·15-s + 5·17-s − i·19-s + 11.1i·21-s + 0.792·23-s − 1.37·25-s − 18.1i·27-s + 2.67i·29-s + 3.46·31-s + ⋯ |
| L(s) = 1 | + 1.94i·3-s − 1.12i·5-s + 1.25·7-s − 2.79·9-s − 0.715i·11-s − 1.61i·13-s + 2.19·15-s + 1.21·17-s − 0.229i·19-s + 2.44i·21-s + 0.165·23-s − 0.274·25-s − 3.48i·27-s + 0.496i·29-s + 0.622·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.762880615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.762880615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + iT \) |
| good | 3 | \( 1 - 3.37iT - 3T^{2} \) |
| 5 | \( 1 + 2.52iT - 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 + 2.37iT - 11T^{2} \) |
| 13 | \( 1 + 5.84iT - 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 23 | \( 1 - 0.792T + 23T^{2} \) |
| 29 | \( 1 - 2.67iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 10.0iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 3.62iT - 43T^{2} \) |
| 47 | \( 1 + 0.644T + 47T^{2} \) |
| 53 | \( 1 + 6.13iT - 53T^{2} \) |
| 59 | \( 1 + 1.37iT - 59T^{2} \) |
| 61 | \( 1 - 14.5iT - 61T^{2} \) |
| 67 | \( 1 + 2.62iT - 67T^{2} \) |
| 71 | \( 1 + 6.63T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 0.294T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 2.74T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.793453424978831950639587956594, −8.935056255208860080553747564614, −8.364372323756952524397053718469, −7.83517240208153323611220012322, −5.72853265142783018441458397021, −5.34130589979314354954984965027, −4.74988223034444358934305836803, −3.80768512251790735407297090530, −2.87994764277527998292431646928, −0.843841788757877423698514414155,
1.39862032517824890457937583829, 2.07375692236326742150565582249, 3.09818265632265382456473813380, 4.65104966035966287521323056743, 5.82186832003373871714413879937, 6.69548887093013696639102278868, 7.13710281756055274267487485458, 7.88286956828739909512687145111, 8.467105976325673024627175568916, 9.656744556135019869121598832131
