| L(s) = 1 | + 4·2-s + 27.8·3-s + 16·4-s + 111.·6-s + 240.·7-s + 64·8-s + 534.·9-s + 544.·11-s + 446.·12-s − 169·13-s + 961.·14-s + 256·16-s − 1.62e3·17-s + 2.13e3·18-s − 805.·19-s + 6.70e3·21-s + 2.17e3·22-s − 373.·23-s + 1.78e3·24-s − 676·26-s + 8.14e3·27-s + 3.84e3·28-s + 1.50e3·29-s − 2.20e3·31-s + 1.02e3·32-s + 1.51e4·33-s − 6.51e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.78·3-s + 0.5·4-s + 1.26·6-s + 1.85·7-s + 0.353·8-s + 2.20·9-s + 1.35·11-s + 0.894·12-s − 0.277·13-s + 1.31·14-s + 0.250·16-s − 1.36·17-s + 1.55·18-s − 0.512·19-s + 3.31·21-s + 0.958·22-s − 0.147·23-s + 0.632·24-s − 0.196·26-s + 2.14·27-s + 0.927·28-s + 0.332·29-s − 0.411·31-s + 0.176·32-s + 2.42·33-s − 0.966·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.448124497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.448124497\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 169T \) |
| good | 3 | \( 1 - 27.8T + 243T^{2} \) |
| 7 | \( 1 - 240.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 544.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.62e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 805.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 373.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.31e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.70e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.93e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.57e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.03e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.58e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.19e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.78e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.56e4T + 8.58e9T^{2} \) |
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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
−9.493437377566180999564146043846, −8.765670026982283328815422666282, −8.091084382420908790824892733140, −7.33961416325794839205208621935, −6.32779923722187225633952688426, −4.52782093436346387701863180333, −4.42700831901146259938861051574, −3.14965351235742196323327859547, −1.99113021213989685235262906358, −1.52453463465547112685543993256,
1.52453463465547112685543993256, 1.99113021213989685235262906358, 3.14965351235742196323327859547, 4.42700831901146259938861051574, 4.52782093436346387701863180333, 6.32779923722187225633952688426, 7.33961416325794839205208621935, 8.091084382420908790824892733140, 8.765670026982283328815422666282, 9.493437377566180999564146043846
