| L(s) = 1 | + (1.81 + 0.832i)2-s + (2.70 + 1.30i)3-s + (2.61 + 3.02i)4-s + (1.23 + 4.84i)5-s + (3.82 + 4.62i)6-s + (7.10 − 7.10i)7-s + (2.23 + 7.68i)8-s + (5.59 + 7.04i)9-s + (−1.79 + 9.83i)10-s + (−16.9 − 12.3i)11-s + (3.11 + 11.5i)12-s + (0.159 − 1.00i)13-s + (18.8 − 7.00i)14-s + (−2.99 + 14.6i)15-s + (−2.32 + 15.8i)16-s + (−4.70 − 9.22i)17-s + ⋯ |
| L(s) = 1 | + (0.909 + 0.416i)2-s + (0.900 + 0.434i)3-s + (0.653 + 0.756i)4-s + (0.246 + 0.969i)5-s + (0.638 + 0.770i)6-s + (1.01 − 1.01i)7-s + (0.279 + 0.960i)8-s + (0.621 + 0.783i)9-s + (−0.179 + 0.983i)10-s + (−1.54 − 1.11i)11-s + (0.259 + 0.965i)12-s + (0.0122 − 0.0775i)13-s + (1.34 − 0.500i)14-s + (−0.199 + 0.979i)15-s + (−0.145 + 0.989i)16-s + (−0.276 − 0.542i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 - 0.952i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.14646 + 2.30104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.14646 + 2.30104i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.81 - 0.832i)T \) |
| 3 | \( 1 + (-2.70 - 1.30i)T \) |
| 5 | \( 1 + (-1.23 - 4.84i)T \) |
| good | 7 | \( 1 + (-7.10 + 7.10i)T - 49iT^{2} \) |
| 11 | \( 1 + (16.9 + 12.3i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-0.159 + 1.00i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (4.70 + 9.22i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-8.42 + 25.9i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (32.9 - 5.21i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-3.21 - 9.90i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-17.9 - 5.81i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (7.49 + 1.18i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-30.4 - 41.8i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (31.4 + 31.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-20.4 + 40.1i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (6.10 - 11.9i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-14.9 - 20.5i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-25.7 - 18.6i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-26.0 - 51.0i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (16.3 + 50.4i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (5.16 - 0.818i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (4.24 + 13.0i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-45.7 - 89.8i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (75.3 + 54.7i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-21.8 + 42.9i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.47693729628203157195440442546, −10.85748015284779666911828398293, −10.12612566207838838971133884414, −8.465166105164321751949764528603, −7.72730786115324182764616321646, −7.02584335949313079409655757664, −5.50895066644861682781432046858, −4.51614081666438222107641311205, −3.30136731938199932964927477888, −2.40171917071639283724466368826,
1.74392994144686132911410881030, 2.36844076215861108299897706002, 4.14240532084103091237546148117, 5.11907868131151579022550367813, 6.04553119200030320416189817450, 7.74907713862591585746535166349, 8.253096155064894083539722475010, 9.570878103089693514981165717010, 10.32381415092926350387744890701, 11.86055959165399722695113721896
