| L(s) = 1 | + 3·3-s − 11·5-s + 15·7-s + 9·9-s + 29·11-s − 82·13-s − 33·15-s + 27·17-s + 19·19-s + 45·21-s − 100·23-s − 4·25-s + 27·27-s − 118·29-s − 70·31-s + 87·33-s − 165·35-s + 232·37-s − 246·39-s + 8·41-s + 287·43-s − 99·45-s − 385·47-s − 118·49-s + 81·51-s + 538·53-s − 319·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.983·5-s + 0.809·7-s + 1/3·9-s + 0.794·11-s − 1.74·13-s − 0.568·15-s + 0.385·17-s + 0.229·19-s + 0.467·21-s − 0.906·23-s − 0.0319·25-s + 0.192·27-s − 0.755·29-s − 0.405·31-s + 0.458·33-s − 0.796·35-s + 1.03·37-s − 1.01·39-s + 0.0304·41-s + 1.01·43-s − 0.327·45-s − 1.19·47-s − 0.344·49-s + 0.222·51-s + 1.39·53-s − 0.782·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
| good | 5 | \( 1 + 11 T + p^{3} T^{2} \) |
| 7 | \( 1 - 15 T + p^{3} T^{2} \) |
| 11 | \( 1 - 29 T + p^{3} T^{2} \) |
| 13 | \( 1 + 82 T + p^{3} T^{2} \) |
| 17 | \( 1 - 27 T + p^{3} T^{2} \) |
| 23 | \( 1 + 100 T + p^{3} T^{2} \) |
| 29 | \( 1 + 118 T + p^{3} T^{2} \) |
| 31 | \( 1 + 70 T + p^{3} T^{2} \) |
| 37 | \( 1 - 232 T + p^{3} T^{2} \) |
| 41 | \( 1 - 8 T + p^{3} T^{2} \) |
| 43 | \( 1 - 287 T + p^{3} T^{2} \) |
| 47 | \( 1 + 385 T + p^{3} T^{2} \) |
| 53 | \( 1 - 538 T + p^{3} T^{2} \) |
| 59 | \( 1 - 300 T + p^{3} T^{2} \) |
| 61 | \( 1 + 901 T + p^{3} T^{2} \) |
| 67 | \( 1 + 132 T + p^{3} T^{2} \) |
| 71 | \( 1 + 472 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1131 T + p^{3} T^{2} \) |
| 79 | \( 1 - 52 T + p^{3} T^{2} \) |
| 83 | \( 1 + 276 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1302 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1310 T + p^{3} T^{2} \) |
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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
−9.313078682917791067359005142242, −8.269158378702956669704241293040, −7.64114981880911347690880224070, −7.12191941310613742974815274682, −5.70827427968166776942337017442, −4.56384374532902597243922686329, −3.96069171904995037943024196839, −2.74757848302024324327018235278, −1.56883876888276279124814684171, 0,
1.56883876888276279124814684171, 2.74757848302024324327018235278, 3.96069171904995037943024196839, 4.56384374532902597243922686329, 5.70827427968166776942337017442, 7.12191941310613742974815274682, 7.64114981880911347690880224070, 8.269158378702956669704241293040, 9.313078682917791067359005142242
