| L(s) = 1 | + 4·5-s + 11-s − 5·13-s − 4·17-s + 3·19-s − 8·23-s + 11·25-s − 4·29-s + 31-s − 7·37-s + 4·41-s − 43-s + 8·47-s − 12·53-s + 4·55-s + 2·61-s − 20·65-s − 3·67-s + 4·71-s − 11·73-s + 15·79-s + 12·83-s − 16·85-s − 12·89-s + 12·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.301·11-s − 1.38·13-s − 0.970·17-s + 0.688·19-s − 1.66·23-s + 11/5·25-s − 0.742·29-s + 0.179·31-s − 1.15·37-s + 0.624·41-s − 0.152·43-s + 1.16·47-s − 1.64·53-s + 0.539·55-s + 0.256·61-s − 2.48·65-s − 0.366·67-s + 0.474·71-s − 1.28·73-s + 1.68·79-s + 1.31·83-s − 1.73·85-s − 1.27·89-s + 1.23·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.372212311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.372212311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.57725104762003, −12.34383603828245, −11.81466284749177, −11.27740570342482, −10.61635260337876, −10.37133678763241, −9.788891741292289, −9.527615900093904, −9.141615255368186, −8.719659065656423, −7.934748922620450, −7.571264415279902, −6.918498837530651, −6.538355211758554, −6.090523773173425, −5.501310435388457, −5.272296098308207, −4.601329630392804, −4.167260498560964, −3.371163673435874, −2.766114698077052, −2.129184490653518, −1.986420898682861, −1.300699794489017, −0.3783538737720384,
0.3783538737720384, 1.300699794489017, 1.986420898682861, 2.129184490653518, 2.766114698077052, 3.371163673435874, 4.167260498560964, 4.601329630392804, 5.272296098308207, 5.501310435388457, 6.090523773173425, 6.538355211758554, 6.918498837530651, 7.571264415279902, 7.934748922620450, 8.719659065656423, 9.141615255368186, 9.527615900093904, 9.788891741292289, 10.37133678763241, 10.61635260337876, 11.27740570342482, 11.81466284749177, 12.34383603828245, 12.57725104762003
