L(s) = 1 | + (−1.39 + 0.204i)2-s + (−0.814 − 1.52i)3-s + (1.91 − 0.573i)4-s + (2.08 − 2.08i)5-s + (1.45 + 1.97i)6-s − 1.14·7-s + (−2.56 + 1.19i)8-s + (−1.67 + 2.48i)9-s + (−2.48 + 3.34i)10-s + (1.67 + 1.67i)11-s + (−2.43 − 2.46i)12-s + (0.146 − 0.146i)13-s + (1.60 − 0.234i)14-s + (−4.88 − 1.48i)15-s + (3.34 − 2.19i)16-s + 5.59i·17-s + ⋯ |
L(s) = 1 | + (−0.989 + 0.144i)2-s + (−0.470 − 0.882i)3-s + (0.958 − 0.286i)4-s + (0.931 − 0.931i)5-s + (0.592 + 0.805i)6-s − 0.433·7-s + (−0.906 + 0.422i)8-s + (−0.558 + 0.829i)9-s + (−0.787 + 1.05i)10-s + (0.504 + 0.504i)11-s + (−0.703 − 0.710i)12-s + (0.0405 − 0.0405i)13-s + (0.428 − 0.0627i)14-s + (−1.26 − 0.384i)15-s + (0.835 − 0.549i)16-s + 1.35i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.489378 - 0.234233i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.489378 - 0.234233i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.204i)T \) |
| 3 | \( 1 + (0.814 + 1.52i)T \) |
good | 5 | \( 1 + (-2.08 + 2.08i)T - 5iT^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 + (-1.67 - 1.67i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.146 + 0.146i)T - 13iT^{2} \) |
| 17 | \( 1 - 5.59iT - 17T^{2} \) |
| 19 | \( 1 + (-1.48 - 1.48i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.34iT - 23T^{2} \) |
| 29 | \( 1 + (3.51 + 3.51i)T + 29iT^{2} \) |
| 31 | \( 1 - 5.83iT - 31T^{2} \) |
| 37 | \( 1 + (4.83 + 4.83i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.610T + 41T^{2} \) |
| 43 | \( 1 + (1.48 - 1.48i)T - 43iT^{2} \) |
| 47 | \( 1 - 6.41T + 47T^{2} \) |
| 53 | \( 1 + (-0.164 + 0.164i)T - 53iT^{2} \) |
| 59 | \( 1 + (9.05 + 9.05i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4.53 + 4.53i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.635 + 0.635i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.90iT - 71T^{2} \) |
| 73 | \( 1 - 7.07iT - 73T^{2} \) |
| 79 | \( 1 + 9.83iT - 79T^{2} \) |
| 83 | \( 1 + (8.09 - 8.09i)T - 83iT^{2} \) |
| 89 | \( 1 - 0.490T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−16.04152980899198313282517734703, −14.38784783441989261318263004208, −12.93268888283903553826811291185, −12.17940975584803982055138831264, −10.63361851716249816660754198814, −9.404737451782279877588433927233, −8.235920566746012977433715157663, −6.69955058569404092588748504954, −5.63088384160470586525400218939, −1.69578359853725675531896880474,
3.10481947319434072200526019304, 5.83526797377675345309218897203, 6.99389724778086933820593516998, 9.141657938539346332277417531504, 9.839330008068927701384528138533, 10.89996512231204696049612845669, 11.80782650961055644880870632247, 13.76223171002311163171837455596, 15.04570204789720383423722336662, 16.11857758571428158175444325836