| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s + 7-s − 3·8-s + 9-s − 10-s + 4·11-s − 12-s + 14-s − 15-s − 16-s + 17-s + 18-s − 4·19-s + 20-s + 21-s + 4·22-s − 3·24-s + 25-s + 27-s − 28-s − 2·29-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 − 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-s − 0.612·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s − 0.371·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.204184381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.204184381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.83173873438978, −12.19942294528352, −11.95092565543968, −11.40581350068921, −11.08005454356124, −10.23084241967762, −9.925674982098886, −9.451018684338947, −8.802473294674731, −8.576606372785443, −8.231840643740699, −7.615562741016152, −6.966942443007392, −6.597328181399938, −6.055674149245675, −5.534392302886816, −4.869124365838056, −4.436867193078749, −4.103700039945432, −3.598957836854193, −3.115233157523327, −2.545962217157407, −1.775387686205457, −1.202809337074526, −0.4259732942857912,
0.4259732942857912, 1.202809337074526, 1.775387686205457, 2.545962217157407, 3.115233157523327, 3.598957836854193, 4.103700039945432, 4.436867193078749, 4.869124365838056, 5.534392302886816, 6.055674149245675, 6.597328181399938, 6.966942443007392, 7.615562741016152, 8.231840643740699, 8.576606372785443, 8.802473294674731, 9.451018684338947, 9.925674982098886, 10.23084241967762, 11.08005454356124, 11.40581350068921, 11.95092565543968, 12.19942294528352, 12.83173873438978
