| L(s) = 1 | + (−1.34 − 0.424i)2-s + (1.63 + 1.14i)4-s − 0.127i·5-s + (−1.72 − 2.24i)8-s + (−0.0542 + 0.172i)10-s − 3.99i·11-s + 0.891i·13-s + (1.37 + 3.75i)16-s + 5.82i·17-s − 6.31·19-s + (0.146 − 0.209i)20-s + (−1.69 + 5.38i)22-s + 6.60i·23-s + 4.98·25-s + (0.378 − 1.20i)26-s + ⋯ |
| L(s) = 1 | + (−0.953 − 0.300i)2-s + (0.819 + 0.572i)4-s − 0.0572i·5-s + (−0.610 − 0.792i)8-s + (−0.0171 + 0.0545i)10-s − 1.20i·11-s + 0.247i·13-s + (0.344 + 0.938i)16-s + 1.41i·17-s − 1.44·19-s + (0.0327 − 0.0469i)20-s + (−0.361 + 1.14i)22-s + 1.37i·23-s + 0.996·25-s + (0.0742 − 0.235i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8728701773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8728701773\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 + 0.424i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.127iT - 5T^{2} \) |
| 11 | \( 1 + 3.99iT - 11T^{2} \) |
| 13 | \( 1 - 0.891iT - 13T^{2} \) |
| 17 | \( 1 - 5.82iT - 17T^{2} \) |
| 19 | \( 1 + 6.31T + 19T^{2} \) |
| 23 | \( 1 - 6.60iT - 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 8.45T + 31T^{2} \) |
| 37 | \( 1 + 8.67T + 37T^{2} \) |
| 41 | \( 1 - 3.24iT - 41T^{2} \) |
| 43 | \( 1 + 0.881iT - 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 7.53T + 53T^{2} \) |
| 59 | \( 1 - 0.588T + 59T^{2} \) |
| 61 | \( 1 - 1.68iT - 61T^{2} \) |
| 67 | \( 1 + 6.35iT - 67T^{2} \) |
| 71 | \( 1 - 11.3iT - 71T^{2} \) |
| 73 | \( 1 - 15.5iT - 73T^{2} \) |
| 79 | \( 1 + 9.82iT - 79T^{2} \) |
| 83 | \( 1 - 1.48T + 83T^{2} \) |
| 89 | \( 1 - 0.449iT - 89T^{2} \) |
| 97 | \( 1 - 16.2iT - 97T^{2} \) |
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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
−9.247594190345895296700938992837, −8.549680091497160480074676873373, −8.205864710103978442519263891459, −7.10209349358260405916801498081, −6.34270582417326646447602858841, −5.61194400268309391105789830604, −4.12992282631778995686363518876, −3.35239484683322114236885443793, −2.21479551985740209487045202492, −1.07593992070296351955054169667,
0.51389867933724660051545162159, 2.04142325801318312142705500857, 2.81058052129261266852987552706, 4.41388144329091430642290860714, 5.15556768960092807578665746384, 6.32352705667002909927826918420, 6.95137785864524302270320977414, 7.55291296797968127494595699478, 8.617611555687978990529454909450, 8.989146786104780736886718247107
