| L(s) = 1 | + (2.10 + 0.447i)2-s + (1.01 + 1.40i)3-s + (2.39 + 1.06i)4-s + (−1.89 − 1.19i)5-s + (1.50 + 3.40i)6-s + (−0.107 + 0.185i)7-s + (1.08 + 0.786i)8-s + (−0.937 + 2.84i)9-s + (−3.44 − 3.35i)10-s + (0.814 + 0.173i)11-s + (0.936 + 4.44i)12-s + (5.35 − 1.13i)13-s + (−0.308 + 0.343i)14-s + (−0.244 − 3.86i)15-s + (−1.58 − 1.75i)16-s + (−4.59 − 3.33i)17-s + ⋯ |
| L(s) = 1 | + (1.48 + 0.316i)2-s + (0.586 + 0.810i)3-s + (1.19 + 0.533i)4-s + (−0.845 − 0.534i)5-s + (0.615 + 1.38i)6-s + (−0.0405 + 0.0702i)7-s + (0.382 + 0.278i)8-s + (−0.312 + 0.949i)9-s + (−1.08 − 1.06i)10-s + (0.245 + 0.0521i)11-s + (0.270 + 1.28i)12-s + (1.48 − 0.315i)13-s + (−0.0825 + 0.0916i)14-s + (−0.0630 − 0.998i)15-s + (−0.395 − 0.439i)16-s + (−1.11 − 0.809i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.37507 + 1.07798i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.37507 + 1.07798i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.01 - 1.40i)T \) |
| 5 | \( 1 + (1.89 + 1.19i)T \) |
| good | 2 | \( 1 + (-2.10 - 0.447i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (0.107 - 0.185i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.814 - 0.173i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-5.35 + 1.13i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (4.59 + 3.33i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (3.93 + 2.85i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.383 - 0.425i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.0969 - 0.922i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.107 - 1.02i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (2.04 - 6.28i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-9.61 + 2.04i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-2.16 + 3.75i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.270 - 2.57i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (7.56 - 5.49i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (9.56 - 2.03i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-14.7 - 3.13i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (1.12 - 10.7i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (2.48 - 1.80i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.33 - 4.10i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.869 + 8.27i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (3.17 - 1.41i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (2.45 + 7.54i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.06 + 10.1i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.71402306933542163897872989560, −11.49279258656865708259954226988, −10.86896695446160898456299052319, −9.175081435038938691423567644847, −8.515161188676313590200290410199, −7.16980610755022199789783312546, −5.87193113080980359132027466344, −4.62544007598209858971730743386, −4.07196259693106199393775863402, −2.92084570496842401514652746858,
2.13328443864075057126307409750, 3.58191266634263940093198417064, 4.14237762546584046497621935566, 6.12841978404699464040132397166, 6.62690861996511198893296476114, 8.044493081460646699804994742694, 8.889063979448834963214881902139, 10.84497675154908275935576738537, 11.37353846323086792481975100627, 12.44443457005168842791118750300
