| L(s) = 1 | + i·3-s + (−1 − i)5-s + 7-s + i·11-s + (1 + i)13-s + (1 − i)15-s + i·21-s + (−1 − i)23-s + i·25-s + i·27-s + (1 − i)29-s − 33-s + (−1 − i)35-s − 37-s + (−1 + i)39-s + ⋯ |
| L(s) = 1 | + i·3-s + (−1 − i)5-s + 7-s + i·11-s + (1 + i)13-s + (1 − i)15-s + i·21-s + (−1 − i)23-s + i·25-s + i·27-s + (1 − i)29-s − 33-s + (−1 − i)35-s − 37-s + (−1 + i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9002647688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9002647688\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - iT - T^{2} \) |
| 5 | \( 1 + (1 + i)T + iT^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + (-1 + i)T - iT^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + 2iT - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + iT^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.96149421639954356875857354642, −10.15012320945537141523753158741, −9.167783884972083478582195708796, −8.441761738449095931571917153573, −7.74478806223013408010087939974, −6.48318490951460633515952943493, −4.90200156301383035106187451122, −4.50342398378809309858466152678, −3.81901970678299416210109689077, −1.71318968817895555125802945762,
1.37734038206400251598907796509, 3.00907142664639510718953363818, 3.93801506688777093517911318691, 5.46658850464577451657301981768, 6.47624356292030367001762775083, 7.36345892124478963795941109643, 8.044193945869278241510149521472, 8.552712122585830614589087572637, 10.28019485156898146002303739418, 10.97180778881695455631165945992
