| L(s) = 1 | − 1.34·2-s − 3-s − 0.202·4-s + 1.16·5-s + 1.34·6-s − 4.05·7-s + 2.95·8-s + 9-s − 1.55·10-s + 0.687·11-s + 0.202·12-s − 5.30·13-s + 5.44·14-s − 1.16·15-s − 3.55·16-s − 17-s − 1.34·18-s + 7.04·19-s − 0.235·20-s + 4.05·21-s − 0.921·22-s + 3.83·23-s − 2.95·24-s − 3.65·25-s + 7.11·26-s − 27-s + 0.822·28-s + ⋯ |
| L(s) = 1 | − 0.947·2-s − 0.577·3-s − 0.101·4-s + 0.519·5-s + 0.547·6-s − 1.53·7-s + 1.04·8-s + 0.333·9-s − 0.492·10-s + 0.207·11-s + 0.0584·12-s − 1.47·13-s + 1.45·14-s − 0.299·15-s − 0.888·16-s − 0.242·17-s − 0.315·18-s + 1.61·19-s − 0.0525·20-s + 0.885·21-s − 0.196·22-s + 0.800·23-s − 0.602·24-s − 0.730·25-s + 1.39·26-s − 0.192·27-s + 0.155·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4058984636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4058984636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 5 | \( 1 - 1.16T + 5T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 11 | \( 1 - 0.687T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 19 | \( 1 - 7.04T + 19T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 - 5.25T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 7.73T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 7.75T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 3.00T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 4.29T + 71T^{2} \) |
| 73 | \( 1 + 2.60T + 73T^{2} \) |
| 83 | \( 1 + 5.07T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 7.18T + 97T^{2} \) |
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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
−8.700912213638418874642266594646, −7.55014064422467303637497709480, −7.08735311161983826117754157404, −6.45802879683830796434258834082, −5.40148689152506716562497931067, −4.95212928414549576846557702697, −3.76438200411000391314925742666, −2.85388926001337574323807364799, −1.65734772362740292279101676935, −0.43559735900560952820837771987,
0.43559735900560952820837771987, 1.65734772362740292279101676935, 2.85388926001337574323807364799, 3.76438200411000391314925742666, 4.95212928414549576846557702697, 5.40148689152506716562497931067, 6.45802879683830796434258834082, 7.08735311161983826117754157404, 7.55014064422467303637497709480, 8.700912213638418874642266594646
