| L(s) = 1 | − 2-s + 2.80·3-s + 4-s + 3.55·5-s − 2.80·6-s − 3.59·7-s − 8-s + 4.87·9-s − 3.55·10-s + 6.34·11-s + 2.80·12-s + 4.97·13-s + 3.59·14-s + 9.98·15-s + 16-s + 4.41·17-s − 4.87·18-s + 7.58·19-s + 3.55·20-s − 10.0·21-s − 6.34·22-s − 7.40·23-s − 2.80·24-s + 7.66·25-s − 4.97·26-s + 5.25·27-s − 3.59·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.62·3-s + 0.5·4-s + 1.59·5-s − 1.14·6-s − 1.35·7-s − 0.353·8-s + 1.62·9-s − 1.12·10-s + 1.91·11-s + 0.810·12-s + 1.38·13-s + 0.961·14-s + 2.57·15-s + 0.250·16-s + 1.07·17-s − 1.14·18-s + 1.73·19-s + 0.795·20-s − 2.20·21-s − 1.35·22-s − 1.54·23-s − 0.572·24-s + 1.53·25-s − 0.976·26-s + 1.01·27-s − 0.679·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.090371522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.090371522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 2.80T + 3T^{2} \) |
| 5 | \( 1 - 3.55T + 5T^{2} \) |
| 7 | \( 1 + 3.59T + 7T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 - 4.97T + 13T^{2} \) |
| 17 | \( 1 - 4.41T + 17T^{2} \) |
| 19 | \( 1 - 7.58T + 19T^{2} \) |
| 23 | \( 1 + 7.40T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 + 7.01T + 41T^{2} \) |
| 43 | \( 1 - 1.66T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 - 6.03T + 59T^{2} \) |
| 61 | \( 1 - 2.84T + 61T^{2} \) |
| 67 | \( 1 + 8.52T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 8.12T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + 8.71T + 83T^{2} \) |
| 89 | \( 1 + 6.49T + 89T^{2} \) |
| 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
−8.393549990415234092416765690488, −7.43054061432144768333408672133, −6.73549429143876127880802546422, −6.10935033481637981902880769850, −5.62254083722889455359197272641, −3.83269037788332976780245247061, −3.49854086780557476158526339683, −2.77684194816755407832937306765, −1.63421121132190780831922527380, −1.32582671684575321171063109935,
1.32582671684575321171063109935, 1.63421121132190780831922527380, 2.77684194816755407832937306765, 3.49854086780557476158526339683, 3.83269037788332976780245247061, 5.62254083722889455359197272641, 6.10935033481637981902880769850, 6.73549429143876127880802546422, 7.43054061432144768333408672133, 8.393549990415234092416765690488
