| L(s) = 1 | + i·3-s + 7-s − i·11-s + (−1 − i)17-s + (1 + i)19-s + i·21-s + (1 + i)23-s − i·25-s + i·27-s + (1 − i)29-s + 33-s − i·37-s + i·41-s − 47-s + (1 − i)51-s + ⋯ |
| L(s) = 1 | + i·3-s + 7-s − i·11-s + (−1 − i)17-s + (1 + i)19-s + i·21-s + (1 + i)23-s − i·25-s + i·27-s + (1 − i)29-s + 33-s − i·37-s + i·41-s − 47-s + (1 − i)51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.412292419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.412292419\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 - iT - T^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (1 + i)T + iT^{2} \) |
| 19 | \( 1 + (-1 - i)T + iT^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 + (-1 + i)T - iT^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + (-1 - i)T + iT^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + (1 - i)T - iT^{2} \) |
| 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
−9.399759391855252633706333856466, −8.491819700015361772715755912856, −7.900370309996706162018499090949, −6.98894435881551160721472798823, −5.95603092820150985488980632861, −5.08741503286691734717081810094, −4.56688016590872500800539516412, −3.64962792483432618928501043322, −2.72072498004235112350166369293, −1.27989749731167350728540471447,
1.30072840266942807358522333967, 1.99127747868552607243849692305, 3.12285922571087412391646162365, 4.66324515160619296370860067532, 4.82307415500940645116658424544, 6.18696278455861847731708733936, 7.02776944463005432686777545275, 7.30140143271157373548602627411, 8.298253680705910316295745946624, 8.833974449705970674576557088184
