| L(s) = 1 | + 2-s − 4-s + 2·5-s + 3·7-s − 3·8-s + 2·10-s + 11-s + 6·13-s + 3·14-s − 16-s + 3·17-s − 2·20-s + 22-s + 5·23-s − 25-s + 6·26-s − 3·28-s − 7·29-s − 8·31-s + 5·32-s + 3·34-s + 6·35-s + 10·37-s − 6·40-s − 2·41-s + 11·43-s − 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.13·7-s − 1.06·8-s + 0.632·10-s + 0.301·11-s + 1.66·13-s + 0.801·14-s − 1/4·16-s + 0.727·17-s − 0.447·20-s + 0.213·22-s + 1.04·23-s − 1/5·25-s + 1.17·26-s − 0.566·28-s − 1.29·29-s − 1.43·31-s + 0.883·32-s + 0.514·34-s + 1.01·35-s + 1.64·37-s − 0.948·40-s − 0.312·41-s + 1.67·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.968086530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.968086530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.62299067994835, −14.48572299974736, −13.89752820080789, −13.30353185730809, −12.97886267469207, −12.63816374273673, −11.59129989349755, −11.31055458781733, −10.95168323086804, −10.05398347828465, −9.514395785516078, −9.066911574830044, −8.511947903675518, −8.014076086416335, −7.290889737611016, −6.513794197612639, −5.745436064533883, −5.636281114082134, −5.037165740810158, −4.189280863844745, −3.805301302171353, −3.116626840640993, −2.175096047320492, −1.464186151012192, −0.8087338487398218,
0.8087338487398218, 1.464186151012192, 2.175096047320492, 3.116626840640993, 3.805301302171353, 4.189280863844745, 5.037165740810158, 5.636281114082134, 5.745436064533883, 6.513794197612639, 7.290889737611016, 8.014076086416335, 8.511947903675518, 9.066911574830044, 9.514395785516078, 10.05398347828465, 10.95168323086804, 11.31055458781733, 11.59129989349755, 12.63816374273673, 12.97886267469207, 13.30353185730809, 13.89752820080789, 14.48572299974736, 14.62299067994835
