| L(s) = 1 | + 1.61·2-s + 2.76·3-s + 0.617·4-s + 3.34·5-s + 4.47·6-s − 5.14·7-s − 2.23·8-s + 4.65·9-s + 5.40·10-s − 1.82·11-s + 1.70·12-s + 4.81·13-s − 8.32·14-s + 9.24·15-s − 4.85·16-s − 2.57·17-s + 7.52·18-s − 0.856·19-s + 2.06·20-s − 14.2·21-s − 2.94·22-s + 4.83·23-s − 6.18·24-s + 6.17·25-s + 7.79·26-s + 4.56·27-s − 3.17·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 1.59·3-s + 0.308·4-s + 1.49·5-s + 1.82·6-s − 1.94·7-s − 0.790·8-s + 1.55·9-s + 1.71·10-s − 0.548·11-s + 0.493·12-s + 1.33·13-s − 2.22·14-s + 2.38·15-s − 1.21·16-s − 0.624·17-s + 1.77·18-s − 0.196·19-s + 0.461·20-s − 3.10·21-s − 0.628·22-s + 1.00·23-s − 1.26·24-s + 1.23·25-s + 1.52·26-s + 0.879·27-s − 0.600·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.823493458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.823493458\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 - 2.76T + 3T^{2} \) |
| 5 | \( 1 - 3.34T + 5T^{2} \) |
| 7 | \( 1 + 5.14T + 7T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 19 | \( 1 + 0.856T + 19T^{2} \) |
| 23 | \( 1 - 4.83T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 + 5.44T + 31T^{2} \) |
| 37 | \( 1 + 7.40T + 37T^{2} \) |
| 41 | \( 1 + 4.34T + 41T^{2} \) |
| 43 | \( 1 + 7.82T + 43T^{2} \) |
| 47 | \( 1 - 4.92T + 47T^{2} \) |
| 53 | \( 1 + 1.60T + 53T^{2} \) |
| 59 | \( 1 - 1.63T + 59T^{2} \) |
| 61 | \( 1 - 1.91T + 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−10.67993808187593685599623441980, −9.781447353315560014679237266554, −9.117044596933447474871549277910, −8.656567370007561838518873601396, −6.86904461199573361807567572551, −6.27350121868203034663611875873, −5.27948673619009850651502761213, −3.70223158778527927115533180505, −3.15587992313466484996921673699, −2.21232746787291006976846572654,
2.21232746787291006976846572654, 3.15587992313466484996921673699, 3.70223158778527927115533180505, 5.27948673619009850651502761213, 6.27350121868203034663611875873, 6.86904461199573361807567572551, 8.656567370007561838518873601396, 9.117044596933447474871549277910, 9.781447353315560014679237266554, 10.67993808187593685599623441980
