| L(s) = 1 | − 3-s + 9-s + 2·11-s + 5·13-s + 2·17-s − 3·19-s − 2·23-s − 5·25-s − 27-s − 8·29-s − 31-s − 2·33-s + 5·37-s − 5·39-s − 2·41-s + 7·43-s − 8·47-s − 2·51-s + 2·53-s + 3·57-s − 10·59-s − 2·61-s − 11·67-s + 2·69-s − 12·71-s + 3·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s + 1.38·13-s + 0.485·17-s − 0.688·19-s − 0.417·23-s − 25-s − 0.192·27-s − 1.48·29-s − 0.179·31-s − 0.348·33-s + 0.821·37-s − 0.800·39-s − 0.312·41-s + 1.06·43-s − 1.16·47-s − 0.280·51-s + 0.274·53-s + 0.397·57-s − 1.30·59-s − 0.256·61-s − 1.34·67-s + 0.240·69-s − 1.42·71-s + 0.351·73-s + 0.577·75-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 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.45069432583647424039400982506, −6.41775947912251872798912711009, −6.05847745868255030754741917218, −5.52066857949157663795299325274, −4.44265288591296618928537467069, −3.93716465709458111849850065488, −3.20959128985187465740432361514, −1.93796867020393819840381463253, −1.26081243376670582836012317818, 0,
1.26081243376670582836012317818, 1.93796867020393819840381463253, 3.20959128985187465740432361514, 3.93716465709458111849850065488, 4.44265288591296618928537467069, 5.52066857949157663795299325274, 6.05847745868255030754741917218, 6.41775947912251872798912711009, 7.45069432583647424039400982506
