| L(s) = 1 | + 2.66·2-s + 1.84·3-s + 5.12·4-s + 5-s + 4.92·6-s + 0.877·7-s + 8.33·8-s + 0.406·9-s + 2.66·10-s + 11-s + 9.45·12-s − 1.10·13-s + 2.34·14-s + 1.84·15-s + 11.9·16-s − 5.26·17-s + 1.08·18-s + 1.15·19-s + 5.12·20-s + 1.61·21-s + 2.66·22-s + 1.08·23-s + 15.3·24-s + 25-s − 2.93·26-s − 4.78·27-s + 4.49·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 1.06·3-s + 2.56·4-s + 0.447·5-s + 2.01·6-s + 0.331·7-s + 2.94·8-s + 0.135·9-s + 0.843·10-s + 0.301·11-s + 2.72·12-s − 0.305·13-s + 0.625·14-s + 0.476·15-s + 2.99·16-s − 1.27·17-s + 0.255·18-s + 0.264·19-s + 1.14·20-s + 0.353·21-s + 0.569·22-s + 0.226·23-s + 3.13·24-s + 0.200·25-s − 0.576·26-s − 0.921·27-s + 0.849·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.25739912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.25739912\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 3 | \( 1 - 1.84T + 3T^{2} \) |
| 7 | \( 1 - 0.877T + 7T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 + 5.26T + 17T^{2} \) |
| 19 | \( 1 - 1.15T + 19T^{2} \) |
| 23 | \( 1 - 1.08T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 - 3.80T + 31T^{2} \) |
| 37 | \( 1 - 1.48T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 3.34T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 + 6.99T + 53T^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 - 8.99T + 61T^{2} \) |
| 67 | \( 1 + 6.29T + 67T^{2} \) |
| 71 | \( 1 - 5.97T + 71T^{2} \) |
| 79 | \( 1 + 1.71T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 6.20T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.305135740894903568475833689383, −7.53519514619010597536661345031, −6.67052568370162936698413979140, −6.24005359199289768995506842426, −5.16420798380322468575594015439, −4.70038368046815319989682096456, −3.80680091914166640931117579323, −3.06408133976634759477987106463, −2.38653626136514610368527013841, −1.66373463972223431084694191705,
1.66373463972223431084694191705, 2.38653626136514610368527013841, 3.06408133976634759477987106463, 3.80680091914166640931117579323, 4.70038368046815319989682096456, 5.16420798380322468575594015439, 6.24005359199289768995506842426, 6.67052568370162936698413979140, 7.53519514619010597536661345031, 8.305135740894903568475833689383
