L(s) = 1 | + 0.903·2-s + (−1.72 + 0.188i)3-s − 1.18·4-s + (1.94 + 1.10i)5-s + (−1.55 + 0.169i)6-s − 2.87·8-s + (2.92 − 0.647i)9-s + (1.75 + 0.994i)10-s − 4.06i·11-s + (2.03 − 0.222i)12-s + 2.88·13-s + (−3.55 − 1.52i)15-s − 0.230·16-s + 6.66i·17-s + (2.64 − 0.585i)18-s + 5.70i·19-s + ⋯ |
L(s) = 1 | + 0.638·2-s + (−0.994 + 0.108i)3-s − 0.591·4-s + (0.870 + 0.492i)5-s + (−0.634 + 0.0693i)6-s − 1.01·8-s + (0.976 − 0.215i)9-s + (0.556 + 0.314i)10-s − 1.22i·11-s + (0.588 − 0.0643i)12-s + 0.799·13-s + (−0.918 − 0.394i)15-s − 0.0575·16-s + 1.61i·17-s + (0.623 − 0.137i)18-s + 1.30i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22093 + 0.673180i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22093 + 0.673180i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.72 - 0.188i)T \) |
| 5 | \( 1 + (-1.94 - 1.10i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.903T + 2T^{2} \) |
| 11 | \( 1 + 4.06iT - 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 - 6.66iT - 17T^{2} \) |
| 19 | \( 1 - 5.70iT - 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 - 5.89iT - 29T^{2} \) |
| 31 | \( 1 + 1.87iT - 31T^{2} \) |
| 37 | \( 1 + 1.59iT - 37T^{2} \) |
| 41 | \( 1 - 9.12T + 41T^{2} \) |
| 43 | \( 1 - 7.53iT - 43T^{2} \) |
| 47 | \( 1 - 6.91iT - 47T^{2} \) |
| 53 | \( 1 - 1.51T + 53T^{2} \) |
| 59 | \( 1 + 0.991T + 59T^{2} \) |
| 61 | \( 1 + 6.16iT - 61T^{2} \) |
| 67 | \( 1 + 10.0iT - 67T^{2} \) |
| 71 | \( 1 - 4.81iT - 71T^{2} \) |
| 73 | \( 1 + 0.560T + 73T^{2} \) |
| 79 | \( 1 - 3.79T + 79T^{2} \) |
| 83 | \( 1 + 4.00iT - 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−10.76711702733806061373322544905, −9.808660809201083894672217998375, −8.940525222614105485322355651740, −7.956737709156017875204471224400, −6.32973802844020567242195445970, −6.04862296806615499257364187878, −5.36803652688994289251144299090, −4.09717437533671915007862505779, −3.29102675702782789172746880653, −1.34218883745080480095745909988,
0.78071212717355042395536443366, 2.46607569338126127032676182817, 4.21015215750301770779993251881, 4.91960982918596494997159263218, 5.50613996065507503794445161792, 6.49455637363522219523233914069, 7.32355678411049488845248228785, 8.812166034336102150520445074095, 9.486289922181187083616511300426, 10.14215259670028953302245313986