| L(s) = 1 | + 5-s − 5·7-s + 11-s + 2·13-s − 3·17-s + 7·19-s − 6·23-s + 25-s + 3·29-s + 7·31-s − 5·35-s − 7·37-s − 6·41-s − 8·43-s + 6·47-s + 18·49-s + 3·53-s + 55-s − 6·59-s − 61-s + 2·65-s − 8·67-s + 3·71-s + 2·73-s − 5·77-s + 10·79-s − 6·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.88·7-s + 0.301·11-s + 0.554·13-s − 0.727·17-s + 1.60·19-s − 1.25·23-s + 1/5·25-s + 0.557·29-s + 1.25·31-s − 0.845·35-s − 1.15·37-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 18/7·49-s + 0.412·53-s + 0.134·55-s − 0.781·59-s − 0.128·61-s + 0.248·65-s − 0.977·67-s + 0.356·71-s + 0.234·73-s − 0.569·77-s + 1.12·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−7.28679623732328143638820659995, −6.68294385935230455232165754249, −6.21042665222416483818616706218, −5.61233011660907159039330651056, −4.66747173875626625361119952379, −3.66472920055909490558892392745, −3.21966043834009651135277473266, −2.36117160605577914908632554046, −1.19357019847452605351639567397, 0,
1.19357019847452605351639567397, 2.36117160605577914908632554046, 3.21966043834009651135277473266, 3.66472920055909490558892392745, 4.66747173875626625361119952379, 5.61233011660907159039330651056, 6.21042665222416483818616706218, 6.68294385935230455232165754249, 7.28679623732328143638820659995
