| L(s) = 1 | + 6.96·2-s + 22.6·3-s + 16.4·4-s − 25·5-s + 157.·6-s + 169.·7-s − 108.·8-s + 270.·9-s − 174.·10-s + 373.·12-s − 25.3·13-s + 1.17e3·14-s − 566.·15-s − 1.27e3·16-s + 2.01e3·17-s + 1.88e3·18-s + 773.·19-s − 412.·20-s + 3.83e3·21-s − 541.·23-s − 2.44e3·24-s + 625·25-s − 176.·26-s + 633.·27-s + 2.78e3·28-s + 5.88e3·29-s − 3.94e3·30-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 1.45·3-s + 0.515·4-s − 0.447·5-s + 1.79·6-s + 1.30·7-s − 0.596·8-s + 1.11·9-s − 0.550·10-s + 0.748·12-s − 0.0415·13-s + 1.60·14-s − 0.650·15-s − 1.24·16-s + 1.69·17-s + 1.37·18-s + 0.491·19-s − 0.230·20-s + 1.89·21-s − 0.213·23-s − 0.868·24-s + 0.200·25-s − 0.0511·26-s + 0.167·27-s + 0.671·28-s + 1.29·29-s − 0.800·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.876599561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.876599561\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 6.96T + 32T^{2} \) |
| 3 | \( 1 - 22.6T + 243T^{2} \) |
| 7 | \( 1 - 169.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 25.3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.01e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 773.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 541.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.88e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 915.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.54e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.09e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.77e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.78e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.87e4T + 8.58e9T^{2} \) |
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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
−9.703144090170684433175352267449, −8.852676470672192572612983371397, −7.939860316033248673269686000263, −7.55505727217138371996493853382, −6.00236740346807402904008456137, −4.95531130268212627103075726626, −4.18046044807616754086161375600, −3.28815642239942390571580646310, −2.50276639446408260070475465165, −1.15389065693642128540366040840,
1.15389065693642128540366040840, 2.50276639446408260070475465165, 3.28815642239942390571580646310, 4.18046044807616754086161375600, 4.95531130268212627103075726626, 6.00236740346807402904008456137, 7.55505727217138371996493853382, 7.939860316033248673269686000263, 8.852676470672192572612983371397, 9.703144090170684433175352267449
