| L(s) = 1 | − 3-s + 2·5-s + 9-s − 2·11-s − 4·13-s − 2·15-s + 6·17-s − 8·19-s + 6·23-s − 25-s − 27-s − 10·29-s − 4·31-s + 2·33-s + 6·37-s + 4·39-s − 6·41-s − 4·43-s + 2·45-s − 8·47-s − 6·51-s + 2·53-s − 4·55-s + 8·57-s + 4·59-s − 8·61-s − 8·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.516·15-s + 1.45·17-s − 1.83·19-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.348·33-s + 0.986·37-s + 0.640·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s − 0.840·51-s + 0.274·53-s − 0.539·55-s + 1.05·57-s + 0.520·59-s − 1.02·61-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.631099201560386317062119026253, −7.71592026516063918650723966264, −7.02288599719673663589831070636, −6.14084677712433387704936155208, −5.41526796219077428009526667120, −4.90441675593865689533188061580, −3.70870379307642839783383641799, −2.52498891624783975371956159355, −1.63174134659533958289094579310, 0,
1.63174134659533958289094579310, 2.52498891624783975371956159355, 3.70870379307642839783383641799, 4.90441675593865689533188061580, 5.41526796219077428009526667120, 6.14084677712433387704936155208, 7.02288599719673663589831070636, 7.71592026516063918650723966264, 8.631099201560386317062119026253
