| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s + 9-s − 10-s + 12-s + 6·13-s − 14-s − 15-s − 16-s + 2·17-s − 18-s − 20-s − 21-s + 8·23-s − 3·24-s + 25-s − 6·26-s − 27-s − 28-s + 2·29-s + 30-s − 4·31-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 + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 1.66·23-s − 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37905 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37905 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.744908250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.744908250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.78705233716252, −14.23242758877223, −13.82679818238434, −13.20872466353931, −12.84318453096380, −12.33656378619000, −11.41412378323759, −10.97201992185302, −10.74526676282943, −10.11373691364384, −9.429833445183482, −8.971219833515737, −8.653875472723849, −7.831702875521610, −7.465958104264541, −6.700098899212344, −6.047865240230688, −5.555214787473983, −4.930916066063638, −4.347314730166904, −3.679432959398745, −2.912477876318848, −1.810383533305984, −1.162046485282680, −0.6987666088926710,
0.6987666088926710, 1.162046485282680, 1.810383533305984, 2.912477876318848, 3.679432959398745, 4.347314730166904, 4.930916066063638, 5.555214787473983, 6.047865240230688, 6.700098899212344, 7.465958104264541, 7.831702875521610, 8.653875472723849, 8.971219833515737, 9.429833445183482, 10.11373691364384, 10.74526676282943, 10.97201992185302, 11.41412378323759, 12.33656378619000, 12.84318453096380, 13.20872466353931, 13.82679818238434, 14.23242758877223, 14.78705233716252
