| L(s) = 1 | + 2-s − 3-s + 4-s + 4.16·5-s − 6-s + 7-s + 8-s + 9-s + 4.16·10-s + 5.46·11-s − 12-s − 0.925·13-s + 14-s − 4.16·15-s + 16-s − 6.82·17-s + 18-s − 1.06·19-s + 4.16·20-s − 21-s + 5.46·22-s − 3.76·23-s − 24-s + 12.3·25-s − 0.925·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.86·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.31·10-s + 1.64·11-s − 0.288·12-s − 0.256·13-s + 0.267·14-s − 1.07·15-s + 0.250·16-s − 1.65·17-s + 0.235·18-s − 0.244·19-s + 0.931·20-s − 0.218·21-s + 1.16·22-s − 0.784·23-s − 0.204·24-s + 2.46·25-s − 0.181·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4074 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4074 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.170845154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.170845154\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 5 | \( 1 - 4.16T + 5T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 0.925T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 + 1.06T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 - 7.57T + 29T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 + 2.14T + 37T^{2} \) |
| 41 | \( 1 - 3.68T + 41T^{2} \) |
| 43 | \( 1 - 3.82T + 43T^{2} \) |
| 47 | \( 1 - 7.42T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 3.24T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 5.84T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 6.40T + 83T^{2} \) |
| 89 | \( 1 + 0.895T + 89T^{2} \) |
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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
−8.787342957690862307885842417297, −7.31254707350401757210814297999, −6.55577010157952406409441523349, −6.20177696177167380742583270947, −5.63023849475493501826073115765, −4.57004612549790897104615424983, −4.27509040189302888598481956775, −2.75883320410092282127578373378, −1.97998253318381960532765600085, −1.20722456426904419546489648115,
1.20722456426904419546489648115, 1.97998253318381960532765600085, 2.75883320410092282127578373378, 4.27509040189302888598481956775, 4.57004612549790897104615424983, 5.63023849475493501826073115765, 6.20177696177167380742583270947, 6.55577010157952406409441523349, 7.31254707350401757210814297999, 8.787342957690862307885842417297
