| L(s) = 1 | − 0.732·2-s − 1.46·4-s + 5-s + 4.46·7-s + 2.53·8-s − 0.732·10-s + 3.46·11-s − 3.26·14-s + 1.07·16-s + 6.73·17-s − 5.46·19-s − 1.46·20-s − 2.53·22-s + 0.535·23-s + 25-s − 6.53·28-s + 2.73·29-s − 3.19·31-s − 5.85·32-s − 4.92·34-s + 4.46·35-s + 4·37-s + 4·38-s + 2.53·40-s + 5.26·41-s − 0.267·43-s − 5.07·44-s + ⋯ |
| L(s) = 1 | − 0.517·2-s − 0.732·4-s + 0.447·5-s + 1.68·7-s + 0.896·8-s − 0.231·10-s + 1.04·11-s − 0.873·14-s + 0.267·16-s + 1.63·17-s − 1.25·19-s − 0.327·20-s − 0.540·22-s + 0.111·23-s + 0.200·25-s − 1.23·28-s + 0.507·29-s − 0.574·31-s − 1.03·32-s − 0.845·34-s + 0.754·35-s + 0.657·37-s + 0.648·38-s + 0.400·40-s + 0.822·41-s − 0.0408·43-s − 0.764·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.171077416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.171077416\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.732T + 2T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 - 0.535T + 23T^{2} \) |
| 29 | \( 1 - 2.73T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 5.26T + 41T^{2} \) |
| 43 | \( 1 + 0.267T + 43T^{2} \) |
| 47 | \( 1 - 0.196T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 - 7.26T + 59T^{2} \) |
| 61 | \( 1 - 4.46T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 1.92T + 79T^{2} \) |
| 83 | \( 1 - 2.53T + 83T^{2} \) |
| 89 | \( 1 - 1.26T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−8.105959598388993577227348464424, −7.41454219473621291201422619448, −6.56895657486889932225141127692, −5.55816525561168931559226293402, −5.16644500752031651852289790871, −4.24741069330984388625454755202, −3.84617516818349399025394938466, −2.41622731051027632890153247896, −1.48070005365659940539394053991, −0.932301200775645613889502663689,
0.932301200775645613889502663689, 1.48070005365659940539394053991, 2.41622731051027632890153247896, 3.84617516818349399025394938466, 4.24741069330984388625454755202, 5.16644500752031651852289790871, 5.55816525561168931559226293402, 6.56895657486889932225141127692, 7.41454219473621291201422619448, 8.105959598388993577227348464424
