| L(s) = 1 | − 2.24·2-s + 3.06·4-s + 3.96·5-s − 4.97·7-s − 2.38·8-s − 8.93·10-s − 6.10·11-s − 0.944·13-s + 11.1·14-s − 0.753·16-s − 0.884·17-s − 2.59·19-s + 12.1·20-s + 13.7·22-s + 2.77·23-s + 10.7·25-s + 2.12·26-s − 15.2·28-s + 1.93·29-s − 3.39·31-s + 6.46·32-s + 1.99·34-s − 19.7·35-s + 2.71·37-s + 5.84·38-s − 9.47·40-s − 1.37·41-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 1.53·4-s + 1.77·5-s − 1.88·7-s − 0.843·8-s − 2.82·10-s − 1.84·11-s − 0.262·13-s + 2.99·14-s − 0.188·16-s − 0.214·17-s − 0.596·19-s + 2.71·20-s + 2.92·22-s + 0.577·23-s + 2.15·25-s + 0.416·26-s − 2.87·28-s + 0.360·29-s − 0.609·31-s + 1.14·32-s + 0.341·34-s − 3.33·35-s + 0.445·37-s + 0.948·38-s − 1.49·40-s − 0.214·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5188047806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5188047806\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 547 | \( 1 + T \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 5 | \( 1 - 3.96T + 5T^{2} \) |
| 7 | \( 1 + 4.97T + 7T^{2} \) |
| 11 | \( 1 + 6.10T + 11T^{2} \) |
| 13 | \( 1 + 0.944T + 13T^{2} \) |
| 17 | \( 1 + 0.884T + 17T^{2} \) |
| 19 | \( 1 + 2.59T + 19T^{2} \) |
| 23 | \( 1 - 2.77T + 23T^{2} \) |
| 29 | \( 1 - 1.93T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 + 6.58T + 53T^{2} \) |
| 59 | \( 1 + 0.838T + 59T^{2} \) |
| 61 | \( 1 + 9.71T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 7.51T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 0.398T + 79T^{2} \) |
| 83 | \( 1 + 4.09T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 3.97T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.532732865340389125891322829246, −7.59695277015695564913152195319, −6.82334020445329753520086906261, −6.36184566712163232614665594865, −5.65290260092968259453455021408, −4.83545087909831036114766638695, −3.13056437636014794491724599965, −2.57117824776658426876370889545, −1.86161685360581048940324167013, −0.47273724433970650675392600372,
0.47273724433970650675392600372, 1.86161685360581048940324167013, 2.57117824776658426876370889545, 3.13056437636014794491724599965, 4.83545087909831036114766638695, 5.65290260092968259453455021408, 6.36184566712163232614665594865, 6.82334020445329753520086906261, 7.59695277015695564913152195319, 8.532732865340389125891322829246
