| L(s) = 1 | − 2-s + 3-s + 4-s + 0.701·5-s − 6-s + 5.18·7-s − 8-s + 9-s − 0.701·10-s + 1.24·11-s + 12-s + 6.30·13-s − 5.18·14-s + 0.701·15-s + 16-s − 3.07·17-s − 18-s + 19-s + 0.701·20-s + 5.18·21-s − 1.24·22-s − 4.62·23-s − 24-s − 4.50·25-s − 6.30·26-s + 27-s + 5.18·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.313·5-s − 0.408·6-s + 1.96·7-s − 0.353·8-s + 0.333·9-s − 0.221·10-s + 0.375·11-s + 0.288·12-s + 1.74·13-s − 1.38·14-s + 0.181·15-s + 0.250·16-s − 0.746·17-s − 0.235·18-s + 0.229·19-s + 0.156·20-s + 1.13·21-s − 0.265·22-s − 0.964·23-s − 0.204·24-s − 0.901·25-s − 1.23·26-s + 0.192·27-s + 0.980·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.860295570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.860295570\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 - 0.701T + 5T^{2} \) |
| 7 | \( 1 - 5.18T + 7T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 - 6.30T + 13T^{2} \) |
| 17 | \( 1 + 3.07T + 17T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 - 5.88T + 29T^{2} \) |
| 31 | \( 1 + 4.72T + 31T^{2} \) |
| 37 | \( 1 + 3.54T + 37T^{2} \) |
| 41 | \( 1 + 3.03T + 41T^{2} \) |
| 43 | \( 1 - 6.62T + 43T^{2} \) |
| 47 | \( 1 - 7.00T + 47T^{2} \) |
| 59 | \( 1 - 1.31T + 59T^{2} \) |
| 61 | \( 1 - 4.84T + 61T^{2} \) |
| 67 | \( 1 + 2.94T + 67T^{2} \) |
| 71 | \( 1 - 2.79T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−8.245015873298281737170249979964, −7.65284788051858646056808515452, −6.83078175467292269943672950734, −5.99338541736231470724581425572, −5.30791619264809400092782175822, −4.24701479953988015411207921772, −3.74716107574584240949584469733, −2.38566389716609226864993450873, −1.75032207402552800836866178556, −1.07047007205596437126723393924,
1.07047007205596437126723393924, 1.75032207402552800836866178556, 2.38566389716609226864993450873, 3.74716107574584240949584469733, 4.24701479953988015411207921772, 5.30791619264809400092782175822, 5.99338541736231470724581425572, 6.83078175467292269943672950734, 7.65284788051858646056808515452, 8.245015873298281737170249979964
