| L(s) = 1 | − 1.33·2-s − 2.45·3-s − 0.225·4-s − 1.10·5-s + 3.27·6-s − 4.00·7-s + 2.96·8-s + 3.03·9-s + 1.47·10-s + 0.233·11-s + 0.553·12-s + 3.10·13-s + 5.33·14-s + 2.71·15-s − 3.49·16-s + 6.85·17-s − 4.04·18-s − 19-s + 0.249·20-s + 9.84·21-s − 0.311·22-s − 5.50·23-s − 7.28·24-s − 3.77·25-s − 4.13·26-s − 0.0920·27-s + 0.901·28-s + ⋯ |
| L(s) = 1 | − 0.942·2-s − 1.41·3-s − 0.112·4-s − 0.494·5-s + 1.33·6-s − 1.51·7-s + 1.04·8-s + 1.01·9-s + 0.466·10-s + 0.0704·11-s + 0.159·12-s + 0.861·13-s + 1.42·14-s + 0.701·15-s − 0.874·16-s + 1.66·17-s − 0.953·18-s − 0.229·19-s + 0.0556·20-s + 2.14·21-s − 0.0663·22-s − 1.14·23-s − 1.48·24-s − 0.755·25-s − 0.811·26-s − 0.0177·27-s + 0.170·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4082918871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4082918871\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 + 1.33T + 2T^{2} \) |
| 3 | \( 1 + 2.45T + 3T^{2} \) |
| 5 | \( 1 + 1.10T + 5T^{2} \) |
| 7 | \( 1 + 4.00T + 7T^{2} \) |
| 11 | \( 1 - 0.233T + 11T^{2} \) |
| 13 | \( 1 - 3.10T + 13T^{2} \) |
| 17 | \( 1 - 6.85T + 17T^{2} \) |
| 23 | \( 1 + 5.50T + 23T^{2} \) |
| 29 | \( 1 - 4.55T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 1.48T + 37T^{2} \) |
| 41 | \( 1 - 8.86T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 2.50T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 9.54T + 59T^{2} \) |
| 61 | \( 1 + 8.44T + 61T^{2} \) |
| 67 | \( 1 - 6.23T + 67T^{2} \) |
| 71 | \( 1 + 0.175T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 6.13T + 79T^{2} \) |
| 83 | \( 1 + 3.96T + 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 + 6.24T + 97T^{2} \) |
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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
−8.047121497474403857967608207024, −7.44337413797908011774105365175, −6.59717200637765617473746186620, −5.98035220045954958937238063843, −5.54629797939831192017431310456, −4.30281022406556237048707542871, −3.89560551886784895110246760639, −2.75054240997362795495753981360, −1.10859537131191356617350586462, −0.52745942135804602194064066262,
0.52745942135804602194064066262, 1.10859537131191356617350586462, 2.75054240997362795495753981360, 3.89560551886784895110246760639, 4.30281022406556237048707542871, 5.54629797939831192017431310456, 5.98035220045954958937238063843, 6.59717200637765617473746186620, 7.44337413797908011774105365175, 8.047121497474403857967608207024
