| L(s) = 1 | − 2·3-s + 4·5-s − 7-s + 9-s − 4·11-s − 4·13-s − 8·15-s − 6·17-s − 2·19-s + 2·21-s − 23-s + 11·25-s + 4·27-s + 6·29-s − 8·31-s + 8·33-s − 4·35-s − 10·37-s + 8·39-s + 6·41-s − 12·43-s + 4·45-s − 8·47-s + 49-s + 12·51-s + 2·53-s − 16·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 2.06·15-s − 1.45·17-s − 0.458·19-s + 0.436·21-s − 0.208·23-s + 11/5·25-s + 0.769·27-s + 1.11·29-s − 1.43·31-s + 1.39·33-s − 0.676·35-s − 1.64·37-s + 1.28·39-s + 0.937·41-s − 1.82·43-s + 0.596·45-s − 1.16·47-s + 1/7·49-s + 1.68·51-s + 0.274·53-s − 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7790913405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7790913405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.77189518175945, −16.14888090781012, −15.67042685968009, −14.76223221184689, −14.31578011375882, −13.53904277667516, −13.04136685799007, −12.74966633150979, −12.00134088378267, −11.24699596517115, −10.64149769776967, −10.13662064621562, −9.884915949099752, −8.911941807173745, −8.511765923993644, −7.284978689495105, −6.716458891782044, −6.252302893636411, −5.547821197319770, −5.071700649220687, −4.651630461559484, −3.198012833660136, −2.324480867427205, −1.893354316761040, −0.4111692734705891,
0.4111692734705891, 1.893354316761040, 2.324480867427205, 3.198012833660136, 4.651630461559484, 5.071700649220687, 5.547821197319770, 6.252302893636411, 6.716458891782044, 7.284978689495105, 8.511765923993644, 8.911941807173745, 9.884915949099752, 10.13662064621562, 10.64149769776967, 11.24699596517115, 12.00134088378267, 12.74966633150979, 13.04136685799007, 13.53904277667516, 14.31578011375882, 14.76223221184689, 15.67042685968009, 16.14888090781012, 16.77189518175945
