| L(s) = 1 | − 3-s + i·4-s + i·5-s + 9-s − i·11-s − i·12-s − i·15-s − 16-s − 20-s + (1 − i)23-s − 25-s − 27-s + i·33-s + i·36-s + (1 − i)37-s + ⋯ |
| L(s) = 1 | − 3-s + i·4-s + i·5-s + 9-s − i·11-s − i·12-s − i·15-s − 16-s − 20-s + (1 − i)23-s − 25-s − 27-s + i·33-s + i·36-s + (1 − i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5124325930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5124325930\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| good | 2 | \( 1 - iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-1 - i)T + iT^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 + 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−13.04626153620637147925560241206, −12.14520824230585135033675814288, −11.16066340036910961031208830099, −10.68830450870167606505493878184, −9.206166716883822325673110415413, −7.83768670628207218172432513277, −6.88508674445321842543244130273, −5.92208702387101527522442263091, −4.28989820608368591912189901280, −2.92269017277789166739562281737,
1.47786184727082608092675371664, 4.50569616635841001221954384387, 5.22744274393448764590226605608, 6.29718563946434510881114313508, 7.52181963981657335249948391031, 9.212166649138198661442524128971, 9.886113795338133613131479961396, 10.95760162931962160367338185741, 11.89286158380736589932985726108, 12.80673196513357283655651651307
