| L(s) = 1 | + 2.56·2-s + 4.56·4-s + 5-s − 0.438·7-s + 6.56·8-s + 2.56·10-s − 1.56·11-s − 13-s − 1.12·14-s + 7.68·16-s + 1.56·17-s − 5.12·19-s + 4.56·20-s − 4·22-s + 2.43·23-s + 25-s − 2.56·26-s − 2·28-s − 7.12·29-s + 6·31-s + 6.56·32-s + 4·34-s − 0.438·35-s − 10.6·37-s − 13.1·38-s + 6.56·40-s + 3.56·41-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 2.28·4-s + 0.447·5-s − 0.165·7-s + 2.31·8-s + 0.810·10-s − 0.470·11-s − 0.277·13-s − 0.300·14-s + 1.92·16-s + 0.378·17-s − 1.17·19-s + 1.01·20-s − 0.852·22-s + 0.508·23-s + 0.200·25-s − 0.502·26-s − 0.377·28-s − 1.32·29-s + 1.07·31-s + 1.15·32-s + 0.685·34-s − 0.0741·35-s − 1.75·37-s − 2.12·38-s + 1.03·40-s + 0.556·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.194333390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.194333390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 7 | \( 1 + 0.438T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 4.68T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 4.68T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−10.88646095615845168123369960683, −10.23759678723276250291360854061, −8.960271906157408174203070085830, −7.63962013293101031886046918865, −6.74099003219979492542022846742, −5.88757229265970843563261417573, −5.12356601790988876342595548419, −4.16458282532561182364840712696, −3.05987472392565070404397409404, −2.03545650757098222040794671648,
2.03545650757098222040794671648, 3.05987472392565070404397409404, 4.16458282532561182364840712696, 5.12356601790988876342595548419, 5.88757229265970843563261417573, 6.74099003219979492542022846742, 7.63962013293101031886046918865, 8.960271906157408174203070085830, 10.23759678723276250291360854061, 10.88646095615845168123369960683
