| L(s) = 1 | − 2-s + 3-s + 4-s + 2.83·5-s − 6-s − 0.0272·7-s − 8-s + 9-s − 2.83·10-s + 2.84·11-s + 12-s − 3.24·13-s + 0.0272·14-s + 2.83·15-s + 16-s + 17-s − 18-s + 4.41·19-s + 2.83·20-s − 0.0272·21-s − 2.84·22-s − 1.50·23-s − 24-s + 3.05·25-s + 3.24·26-s + 27-s − 0.0272·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.26·5-s − 0.408·6-s − 0.0102·7-s − 0.353·8-s + 0.333·9-s − 0.897·10-s + 0.858·11-s + 0.288·12-s − 0.899·13-s + 0.00727·14-s + 0.732·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.01·19-s + 0.634·20-s − 0.00594·21-s − 0.607·22-s − 0.313·23-s − 0.204·24-s + 0.611·25-s + 0.636·26-s + 0.192·27-s − 0.00514·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.550177413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.550177413\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 + 0.0272T + 7T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 + 3.24T + 13T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 + 1.50T + 23T^{2} \) |
| 29 | \( 1 + 3.74T + 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 - 6.64T + 37T^{2} \) |
| 41 | \( 1 - 3.94T + 41T^{2} \) |
| 43 | \( 1 - 4.59T + 43T^{2} \) |
| 47 | \( 1 + 6.12T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 61 | \( 1 + 8.31T + 61T^{2} \) |
| 67 | \( 1 - 5.55T + 67T^{2} \) |
| 71 | \( 1 + 4.30T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 2.97T + 79T^{2} \) |
| 83 | \( 1 - 2.61T + 83T^{2} \) |
| 89 | \( 1 - 7.11T + 89T^{2} \) |
| 97 | \( 1 + 3.04T + 97T^{2} \) |
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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
−8.049905211811901912410121238082, −7.51390718691820401499552396175, −6.73941624109105307913713699513, −6.07393425305945809629967798617, −5.38260040233368540175607056331, −4.43686286169250074930010807230, −3.38523266077949911214326924425, −2.53351528635618893664108480892, −1.85534371369605523467758374725, −0.948113362683230798741488401762,
0.948113362683230798741488401762, 1.85534371369605523467758374725, 2.53351528635618893664108480892, 3.38523266077949911214326924425, 4.43686286169250074930010807230, 5.38260040233368540175607056331, 6.07393425305945809629967798617, 6.73941624109105307913713699513, 7.51390718691820401499552396175, 8.049905211811901912410121238082
