L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3·7-s + 8-s + 9-s + 10-s − 12-s + 3·13-s + 3·14-s − 15-s + 16-s − 17-s + 18-s − 19-s + 20-s − 3·21-s + 6·23-s − 24-s + 25-s + 3·26-s − 27-s + 3·28-s + 5·29-s − 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.832·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.654·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.192·27-s + 0.566·28-s + 0.928·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.407390458\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.407390458\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.415435677397605243936424049877, −7.78467683022267568961147940653, −6.68457089562479975135270276274, −6.38341047131041426907493112893, −5.25353730936904146672454227745, −4.98431004176741669475048395866, −4.09995499582563850024672271705, −3.07677522642405354590910684711, −1.94478798808142256240066283352, −1.09802099532566774007061542404,
1.09802099532566774007061542404, 1.94478798808142256240066283352, 3.07677522642405354590910684711, 4.09995499582563850024672271705, 4.98431004176741669475048395866, 5.25353730936904146672454227745, 6.38341047131041426907493112893, 6.68457089562479975135270276274, 7.78467683022267568961147940653, 8.415435677397605243936424049877