| L(s) = 1 | − 2-s + 3-s + 4-s − 2.60·5-s − 6-s − 4.26·7-s − 8-s + 9-s + 2.60·10-s + 3.88·11-s + 12-s + 1.17·13-s + 4.26·14-s − 2.60·15-s + 16-s + 7.34·17-s − 18-s + 19-s − 2.60·20-s − 4.26·21-s − 3.88·22-s − 3.84·23-s − 24-s + 1.79·25-s − 1.17·26-s + 27-s − 4.26·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.16·5-s − 0.408·6-s − 1.61·7-s − 0.353·8-s + 0.333·9-s + 0.824·10-s + 1.17·11-s + 0.288·12-s + 0.324·13-s + 1.13·14-s − 0.673·15-s + 0.250·16-s + 1.78·17-s − 0.235·18-s + 0.229·19-s − 0.582·20-s − 0.929·21-s − 0.827·22-s − 0.802·23-s − 0.204·24-s + 0.359·25-s − 0.229·26-s + 0.192·27-s − 0.805·28-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
| \(L(1)\) |
\(\approx\) |
\(1.021766430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.021766430\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.60T + 5T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 23 | \( 1 + 3.84T + 23T^{2} \) |
| 29 | \( 1 - 6.30T + 29T^{2} \) |
| 31 | \( 1 + 6.13T + 31T^{2} \) |
| 37 | \( 1 + 7.87T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 - 5.84T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 59 | \( 1 - 4.82T + 59T^{2} \) |
| 61 | \( 1 - 5.71T + 61T^{2} \) |
| 67 | \( 1 - 4.23T + 67T^{2} \) |
| 71 | \( 1 - 4.20T + 71T^{2} \) |
| 73 | \( 1 - 1.42T + 73T^{2} \) |
| 79 | \( 1 + 0.672T + 79T^{2} \) |
| 83 | \( 1 + 1.22T + 83T^{2} \) |
| 89 | \( 1 + 3.71T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.220951590845407946458164967295, −7.40499956006422740004217361085, −6.85060812296437016844880859982, −6.27723031113960558058112351982, −5.31155028730293854481521123713, −3.92451872087479909411371341961, −3.54350306271556157118584248240, −3.06268981939312302980307241796, −1.67397201222874965821551742102, −0.58068759326784348972685782508,
0.58068759326784348972685782508, 1.67397201222874965821551742102, 3.06268981939312302980307241796, 3.54350306271556157118584248240, 3.92451872087479909411371341961, 5.31155028730293854481521123713, 6.27723031113960558058112351982, 6.85060812296437016844880859982, 7.40499956006422740004217361085, 8.220951590845407946458164967295
