| L(s) = 1 | − 2-s + 4-s + 5-s − 5·7-s − 8-s − 10-s + 5·11-s − 13-s + 5·14-s + 16-s − 6·17-s − 6·19-s + 20-s − 5·22-s + 4·23-s − 4·25-s + 26-s − 5·28-s − 2·29-s + 31-s − 32-s + 6·34-s − 5·35-s − 6·37-s + 6·38-s − 40-s − 12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.88·7-s − 0.353·8-s − 0.316·10-s + 1.50·11-s − 0.277·13-s + 1.33·14-s + 1/4·16-s − 1.45·17-s − 1.37·19-s + 0.223·20-s − 1.06·22-s + 0.834·23-s − 4/5·25-s + 0.196·26-s − 0.944·28-s − 0.371·29-s + 0.179·31-s − 0.176·32-s + 1.02·34-s − 0.845·35-s − 0.986·37-s + 0.973·38-s − 0.158·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.897800505878514138826918489618, −9.064499359203391672641516805909, −8.799134072098402372739077096433, −7.10065667097095096519189382958, −6.57913304235760950474733326879, −5.99142080412013975847311847942, −4.26906324020580354961834542498, −3.19209557202869156366521081245, −1.92254093563913484185105860470, 0,
1.92254093563913484185105860470, 3.19209557202869156366521081245, 4.26906324020580354961834542498, 5.99142080412013975847311847942, 6.57913304235760950474733326879, 7.10065667097095096519189382958, 8.799134072098402372739077096433, 9.064499359203391672641516805909, 9.897800505878514138826918489618
