| L(s) = 1 | + 5-s + 2·7-s − 11-s − 4·13-s − 4·17-s − 4·23-s + 25-s + 6·29-s + 8·31-s + 2·35-s − 2·37-s + 10·41-s − 10·43-s − 12·47-s − 3·49-s − 6·53-s − 55-s − 12·59-s + 10·61-s − 4·65-s + 8·67-s + 4·73-s − 2·77-s − 4·79-s + 2·83-s − 4·85-s + 10·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.301·11-s − 1.10·13-s − 0.970·17-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s + 1.56·41-s − 1.52·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.134·55-s − 1.56·59-s + 1.28·61-s − 0.496·65-s + 0.977·67-s + 0.468·73-s − 0.227·77-s − 0.450·79-s + 0.219·83-s − 0.433·85-s + 1.05·89-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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.67333110756866329874416401978, −6.57327748240581171818887854545, −6.36766165459594369654075552405, −5.08048522522678773409198669875, −4.91305922288279475614074220132, −4.07031386928881137767858771841, −2.87370588881076089291455946719, −2.27520128410229838778949893418, −1.38512320018736524486240508912, 0,
1.38512320018736524486240508912, 2.27520128410229838778949893418, 2.87370588881076089291455946719, 4.07031386928881137767858771841, 4.91305922288279475614074220132, 5.08048522522678773409198669875, 6.36766165459594369654075552405, 6.57327748240581171818887854545, 7.67333110756866329874416401978
