| L(s) = 1 | + 1.45·2-s − 3-s + 0.104·4-s + 5-s − 1.45·6-s + 1.14·7-s − 2.74·8-s + 9-s + 1.45·10-s + 4.77·11-s − 0.104·12-s + 4.54·13-s + 1.65·14-s − 15-s − 4.19·16-s − 7.04·17-s + 1.45·18-s + 1.20·19-s + 0.104·20-s − 1.14·21-s + 6.92·22-s − 3.46·23-s + 2.74·24-s + 25-s + 6.59·26-s − 27-s + 0.119·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s − 0.577·3-s + 0.0524·4-s + 0.447·5-s − 0.592·6-s + 0.430·7-s − 0.972·8-s + 0.333·9-s + 0.458·10-s + 1.43·11-s − 0.0302·12-s + 1.26·13-s + 0.442·14-s − 0.258·15-s − 1.04·16-s − 1.70·17-s + 0.341·18-s + 0.276·19-s + 0.0234·20-s − 0.248·21-s + 1.47·22-s − 0.721·23-s + 0.561·24-s + 0.200·25-s + 1.29·26-s − 0.192·27-s + 0.0226·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.481343713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.481343713\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 1.45T + 2T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 17 | \( 1 + 7.04T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6.62T + 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 - 8.28T + 37T^{2} \) |
| 41 | \( 1 - 9.99T + 41T^{2} \) |
| 43 | \( 1 - 0.0734T + 43T^{2} \) |
| 47 | \( 1 - 6.75T + 47T^{2} \) |
| 53 | \( 1 + 1.65T + 53T^{2} \) |
| 59 | \( 1 - 8.67T + 59T^{2} \) |
| 61 | \( 1 - 3.58T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 83 | \( 1 + 3.91T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 4.82T + 97T^{2} \) |
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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
−9.712973321704860150475592638789, −8.979156077466330198527631607679, −8.273180172098932522702833482652, −6.67777954371259024359277800233, −6.32228355223505466402780739705, −5.55953001001376404162724151892, −4.27264136140503737217427096122, −4.19962041425500941334228583469, −2.63090264502148078070676221643, −1.14651071817493826312681180457,
1.14651071817493826312681180457, 2.63090264502148078070676221643, 4.19962041425500941334228583469, 4.27264136140503737217427096122, 5.55953001001376404162724151892, 6.32228355223505466402780739705, 6.67777954371259024359277800233, 8.273180172098932522702833482652, 8.979156077466330198527631607679, 9.712973321704860150475592638789
