L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 3·9-s − 10-s + 4·11-s − 6·13-s − 14-s + 16-s + 2·17-s − 3·18-s − 20-s + 4·22-s + 25-s − 6·26-s − 28-s + 6·29-s + 8·31-s + 32-s + 2·34-s + 35-s − 3·36-s − 10·37-s − 40-s + 2·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 9-s − 0.316·10-s + 1.20·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.223·20-s + 0.852·22-s + 1/5·25-s − 1.17·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.169·35-s − 1/2·36-s − 1.64·37-s − 0.158·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.180304346\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.180304346\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−14.53586643730421316812320958185, −13.89417772743578510110013468150, −12.14707369305424687183170289720, −11.96572717133546281866330503535, −10.37599431086570880625729241811, −8.977646092372605556420891081568, −7.45420248729985818641828193491, −6.17616144092315208416665412995, −4.63923341949934829303628265619, −3.00381372509091040158022668795,
3.00381372509091040158022668795, 4.63923341949934829303628265619, 6.17616144092315208416665412995, 7.45420248729985818641828193491, 8.977646092372605556420891081568, 10.37599431086570880625729241811, 11.96572717133546281866330503535, 12.14707369305424687183170289720, 13.89417772743578510110013468150, 14.53586643730421316812320958185