| L(s) = 1 | − 3-s + 2·5-s + 9-s + 2·11-s − 13-s − 2·15-s − 19-s − 25-s − 27-s − 4·29-s + 9·31-s − 2·33-s − 3·37-s + 39-s − 10·41-s − 5·43-s + 2·45-s − 6·47-s − 12·53-s + 4·55-s + 57-s + 12·59-s − 10·61-s − 2·65-s + 5·67-s − 6·71-s − 3·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.516·15-s − 0.229·19-s − 1/5·25-s − 0.192·27-s − 0.742·29-s + 1.61·31-s − 0.348·33-s − 0.493·37-s + 0.160·39-s − 1.56·41-s − 0.762·43-s + 0.298·45-s − 0.875·47-s − 1.64·53-s + 0.539·55-s + 0.132·57-s + 1.56·59-s − 1.28·61-s − 0.248·65-s + 0.610·67-s − 0.712·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.18600657975963865110687099932, −6.47163678016630915579213924785, −6.16105890262290710356343811506, −5.26662810397748876822536107657, −4.78653021196099663078132268177, −3.88487180477739169303803683021, −3.01571180501218165451151256437, −1.98158362433052518750362144050, −1.34413459646525300447507524468, 0,
1.34413459646525300447507524468, 1.98158362433052518750362144050, 3.01571180501218165451151256437, 3.88487180477739169303803683021, 4.78653021196099663078132268177, 5.26662810397748876822536107657, 6.16105890262290710356343811506, 6.47163678016630915579213924785, 7.18600657975963865110687099932
