| L(s) = 1 | + 3-s − 4·5-s + 9-s − 4·11-s − 13-s − 4·15-s + 17-s − 19-s + 2·23-s + 11·25-s + 27-s − 6·29-s − 9·31-s − 4·33-s + 11·37-s − 39-s + 10·41-s + 7·43-s − 4·45-s + 6·47-s + 51-s + 6·53-s + 16·55-s − 57-s − 8·59-s − 6·61-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 1.03·15-s + 0.242·17-s − 0.229·19-s + 0.417·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s − 1.61·31-s − 0.696·33-s + 1.80·37-s − 0.160·39-s + 1.56·41-s + 1.06·43-s − 0.596·45-s + 0.875·47-s + 0.140·51-s + 0.824·53-s + 2.15·55-s − 0.132·57-s − 1.04·59-s − 0.768·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.37296854548519, −12.93756549496944, −12.56233870525140, −12.28171450974482, −11.49639520534105, −11.13566199112188, −10.78003474492560, −10.34803006724795, −9.479322614053074, −9.185669288573607, −8.652349587730801, −8.034185758890815, −7.693483020677134, −7.304337887161577, −7.166578207565536, −6.023653059691776, −5.667666844194506, −4.921889334500774, −4.349551337832394, −4.018763005849694, −3.467400881234200, −2.699587226933139, −2.572152271692983, −1.497961849415312, −0.6534491340569341, 0,
0.6534491340569341, 1.497961849415312, 2.572152271692983, 2.699587226933139, 3.467400881234200, 4.018763005849694, 4.349551337832394, 4.921889334500774, 5.667666844194506, 6.023653059691776, 7.166578207565536, 7.304337887161577, 7.693483020677134, 8.034185758890815, 8.652349587730801, 9.185669288573607, 9.479322614053074, 10.34803006724795, 10.78003474492560, 11.13566199112188, 11.49639520534105, 12.28171450974482, 12.56233870525140, 12.93756549496944, 13.37296854548519
