| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 12-s − 13-s − 14-s − 15-s + 16-s + 6·17-s + 18-s + 4·19-s − 20-s − 21-s + 24-s + 25-s − 26-s + 27-s − 28-s + 6·29-s − 30-s + 8·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.504573902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.504573902\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.62631506516367, −12.03332343598423, −11.93873120743036, −11.48974252060312, −10.71528290540607, −10.25898076287031, −9.955918919054598, −9.534999357446949, −8.844291359841086, −8.339836851949162, −8.031768655062970, −7.375027898224804, −7.111354251509965, −6.613203700921873, −5.981176675314139, −5.444990711797166, −5.057932518087230, −4.480884082799663, −3.844367574459020, −3.545361970388446, −2.929237361087183, −2.621868524842895, −1.909352356533557, −0.9939162361036644, −0.7333506652570860,
0.7333506652570860, 0.9939162361036644, 1.909352356533557, 2.621868524842895, 2.929237361087183, 3.545361970388446, 3.844367574459020, 4.480884082799663, 5.057932518087230, 5.444990711797166, 5.981176675314139, 6.613203700921873, 7.111354251509965, 7.375027898224804, 8.031768655062970, 8.339836851949162, 8.844291359841086, 9.534999357446949, 9.955918919054598, 10.25898076287031, 10.71528290540607, 11.48974252060312, 11.93873120743036, 12.03332343598423, 12.62631506516367
