| L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 5·7-s − 2·10-s + 2·11-s − 10·14-s − 4·16-s − 2·17-s − 2·20-s + 4·22-s − 6·23-s + 25-s − 10·28-s + 4·29-s + 7·31-s − 8·32-s − 4·34-s + 5·35-s + 2·37-s + 6·41-s + 43-s + 4·44-s − 12·46-s − 8·47-s + 18·49-s + 2·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 1.88·7-s − 0.632·10-s + 0.603·11-s − 2.67·14-s − 16-s − 0.485·17-s − 0.447·20-s + 0.852·22-s − 1.25·23-s + 1/5·25-s − 1.88·28-s + 0.742·29-s + 1.25·31-s − 1.41·32-s − 0.685·34-s + 0.845·35-s + 0.328·37-s + 0.937·41-s + 0.152·43-s + 0.603·44-s − 1.76·46-s − 1.16·47-s + 18/7·49-s + 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.343946111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.343946111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−7.63648400631118572945359646491, −6.73575865375625776576514356546, −6.35427712283624353827318585549, −5.95161956431808887312609815473, −4.91171712363332031612140653143, −4.15858585797107076582450382736, −3.68786780690385157619911236441, −2.99588992054775671916569915012, −2.30237073160704896817236742844, −0.58289037737084662814429957393,
0.58289037737084662814429957393, 2.30237073160704896817236742844, 2.99588992054775671916569915012, 3.68786780690385157619911236441, 4.15858585797107076582450382736, 4.91171712363332031612140653143, 5.95161956431808887312609815473, 6.35427712283624353827318585549, 6.73575865375625776576514356546, 7.63648400631118572945359646491
