| L(s) = 1 | + 2.20·2-s − 2.22·3-s + 2.86·4-s − 2.44·5-s − 4.90·6-s + 2.05·7-s + 1.91·8-s + 1.94·9-s − 5.38·10-s − 11-s − 6.38·12-s + 0.865·13-s + 4.53·14-s + 5.42·15-s − 1.50·16-s − 2.71·17-s + 4.29·18-s + 3.51·19-s − 7.00·20-s − 4.57·21-s − 2.20·22-s − 4.26·24-s + 0.954·25-s + 1.91·26-s + 2.34·27-s + 5.89·28-s − 8.65·29-s + ⋯ |
| L(s) = 1 | + 1.56·2-s − 1.28·3-s + 1.43·4-s − 1.09·5-s − 2.00·6-s + 0.776·7-s + 0.678·8-s + 0.648·9-s − 1.70·10-s − 0.301·11-s − 1.84·12-s + 0.240·13-s + 1.21·14-s + 1.40·15-s − 0.376·16-s − 0.659·17-s + 1.01·18-s + 0.805·19-s − 1.56·20-s − 0.997·21-s − 0.470·22-s − 0.870·24-s + 0.190·25-s + 0.374·26-s + 0.451·27-s + 1.11·28-s − 1.60·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.086318027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.086318027\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 3 | \( 1 + 2.22T + 3T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 13 | \( 1 - 0.865T + 13T^{2} \) |
| 17 | \( 1 + 2.71T + 17T^{2} \) |
| 19 | \( 1 - 3.51T + 19T^{2} \) |
| 29 | \( 1 + 8.65T + 29T^{2} \) |
| 31 | \( 1 + 0.779T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 4.60T + 41T^{2} \) |
| 43 | \( 1 - 5.92T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 1.98T + 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 + 8.59T + 61T^{2} \) |
| 67 | \( 1 + 3.35T + 67T^{2} \) |
| 71 | \( 1 - 7.04T + 71T^{2} \) |
| 73 | \( 1 + 5.04T + 73T^{2} \) |
| 79 | \( 1 - 5.55T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 1.27T + 89T^{2} \) |
| 97 | \( 1 - 9.38T + 97T^{2} \) |
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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
−7.65641303042005412393619467486, −7.29624105755683867160028214493, −6.33656130044200502555143514863, −5.73962223300774042673748736862, −5.19643846624283221515654558506, −4.49505935356440655521601723285, −4.03329781652080867838909991858, −3.15762894558965570670067827799, −2.06201128066576807946070576686, −0.62818243673020783932575422918,
0.62818243673020783932575422918, 2.06201128066576807946070576686, 3.15762894558965570670067827799, 4.03329781652080867838909991858, 4.49505935356440655521601723285, 5.19643846624283221515654558506, 5.73962223300774042673748736862, 6.33656130044200502555143514863, 7.29624105755683867160028214493, 7.65641303042005412393619467486
