| L(s) = 1 | + 2-s − 3-s − 4-s − 2.85·5-s − 6-s − 7-s − 3·8-s + 9-s − 2.85·10-s + 12-s + 1.23·13-s − 14-s + 2.85·15-s − 16-s − 7.85·17-s + 18-s − 2.61·19-s + 2.85·20-s + 21-s − 3.09·23-s + 3·24-s + 3.14·25-s + 1.23·26-s − 27-s + 28-s + 2·29-s + 2.85·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.5·4-s − 1.27·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 0.333·9-s − 0.902·10-s + 0.288·12-s + 0.342·13-s − 0.267·14-s + 0.736·15-s − 0.250·16-s − 1.90·17-s + 0.235·18-s − 0.600·19-s + 0.638·20-s + 0.218·21-s − 0.644·23-s + 0.612·24-s + 0.629·25-s + 0.242·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.521·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5321984488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5321984488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 + 2.61T + 19T^{2} \) |
| 23 | \( 1 + 3.09T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 7.32T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 6.76T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 5.23T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 - 1.09T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
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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
−8.816316193648678614747280199707, −8.233263839458746192188372506956, −7.16377255829697943252829319920, −6.53489529752984582730625673075, −5.68291193655437884592204819590, −4.82121280744219104749810978658, −3.98796623431240742392134779001, −3.74790328066747404480917589662, −2.33333190707380805847933193303, −0.40658231179387011003891557820,
0.40658231179387011003891557820, 2.33333190707380805847933193303, 3.74790328066747404480917589662, 3.98796623431240742392134779001, 4.82121280744219104749810978658, 5.68291193655437884592204819590, 6.53489529752984582730625673075, 7.16377255829697943252829319920, 8.233263839458746192188372506956, 8.816316193648678614747280199707
