| L(s) = 1 | − 2·2-s + 8.90·3-s + 4·4-s + 1.30·5-s − 17.8·6-s + 32.9·7-s − 8·8-s + 52.3·9-s − 2.60·10-s + 39.2·11-s + 35.6·12-s + 67.5·13-s − 65.8·14-s + 11.6·15-s + 16·16-s − 104.·18-s − 26.0·19-s + 5.21·20-s + 293.·21-s − 78.4·22-s − 50.0·23-s − 71.2·24-s − 123.·25-s − 135.·26-s + 225.·27-s + 131.·28-s − 215.·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.71·3-s + 0.5·4-s + 0.116·5-s − 1.21·6-s + 1.77·7-s − 0.353·8-s + 1.93·9-s − 0.0823·10-s + 1.07·11-s + 0.857·12-s + 1.44·13-s − 1.25·14-s + 0.199·15-s + 0.250·16-s − 1.37·18-s − 0.314·19-s + 0.0582·20-s + 3.04·21-s − 0.760·22-s − 0.454·23-s − 0.606·24-s − 0.986·25-s − 1.01·26-s + 1.61·27-s + 0.889·28-s − 1.38·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.728497912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.728497912\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 8.90T + 27T^{2} \) |
| 5 | \( 1 - 1.30T + 125T^{2} \) |
| 7 | \( 1 - 32.9T + 343T^{2} \) |
| 11 | \( 1 - 39.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 67.5T + 2.19e3T^{2} \) |
| 19 | \( 1 + 26.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 50.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 215.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 93.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 193.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 124.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 202.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 569.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 137.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 3.34T + 2.26e5T^{2} \) |
| 67 | \( 1 + 764.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 492.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 342.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 645.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 830.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 956.T + 9.12e5T^{2} \) |
| 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
−10.09027655969640150591106299956, −8.983826469675624775270938417280, −8.702620820398632460684147013007, −7.924364162731229211649637752502, −7.24256668001439506101172493924, −5.88104347407779040537862764407, −4.28132300048933272850542783814, −3.49118546816396981235068597535, −1.86765980613115057425457922426, −1.53994306632797557318555896158,
1.53994306632797557318555896158, 1.86765980613115057425457922426, 3.49118546816396981235068597535, 4.28132300048933272850542783814, 5.88104347407779040537862764407, 7.24256668001439506101172493924, 7.924364162731229211649637752502, 8.702620820398632460684147013007, 8.983826469675624775270938417280, 10.09027655969640150591106299956
