L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (0.230 + 2.22i)5-s + (−0.866 − 0.499i)6-s + (−0.432 + 0.749i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−1.81 + 1.31i)10-s + (0.151 − 0.0874i)11-s − 0.999i·12-s + (−1.35 + 3.34i)13-s − 0.865·14-s + (−1.31 − 1.81i)15-s + (−0.5 − 0.866i)16-s + (−7.08 − 4.08i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.103 + 0.994i)5-s + (−0.353 − 0.204i)6-s + (−0.163 + 0.283i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.572 + 0.414i)10-s + (0.0456 − 0.0263i)11-s − 0.288i·12-s + (−0.375 + 0.926i)13-s − 0.231·14-s + (−0.338 − 0.467i)15-s + (−0.125 − 0.216i)16-s + (−1.71 − 0.991i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.158118 + 1.06479i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.158118 + 1.06479i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.230 - 2.22i)T \) |
| 13 | \( 1 + (1.35 - 3.34i)T \) |
good | 7 | \( 1 + (0.432 - 0.749i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.151 + 0.0874i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (7.08 + 4.08i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.20 - 3.00i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.52 - 1.45i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.24 - 5.62i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.95iT - 31T^{2} \) |
| 37 | \( 1 + (-0.879 - 1.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.08 - 4.08i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.94 - 4.58i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 2.48iT - 53T^{2} \) |
| 59 | \( 1 + (6.09 + 3.51i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.98 + 6.90i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.36 + 2.36i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.2 - 7.08i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 9.48T + 79T^{2} \) |
| 83 | \( 1 + 0.139T + 83T^{2} \) |
| 89 | \( 1 + (11.3 - 6.56i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.32 + 7.48i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.62481248304269502695483946761, −11.03077797900610063956136943144, −9.765040244887583397792532232413, −9.159763003996106786383828602506, −7.68644700911757678406309813848, −6.82808634598637316244050913804, −6.13692377414886330989178152263, −5.00522997830808365149211542829, −3.90229530435769496877178585078, −2.53267067372415735888785885212,
0.67008528692303149694422390417, 2.26320642899661455887991358574, 3.97119591157888938428026562264, 4.95429994461003166821891841269, 5.83004697902622475560416325551, 6.99866101719093589300629729598, 8.266655011758995178735316298537, 9.159725648501165945248947198059, 10.22899612486075784587306078043, 10.92526056635595335630364033791