| L(s) = 1 | + 0.495·2-s − 1.75·4-s − 0.247·5-s + 1.35·7-s − 1.85·8-s − 0.122·10-s − 4.41·11-s − 0.943·13-s + 0.671·14-s + 2.58·16-s + 1.37·19-s + 0.434·20-s − 2.18·22-s − 2.07·23-s − 4.93·25-s − 0.466·26-s − 2.38·28-s + 2.21·29-s − 9.03·31-s + 5.00·32-s − 0.335·35-s − 2.33·37-s + 0.682·38-s + 0.460·40-s + 9.21·41-s + 3.77·43-s + 7.74·44-s + ⋯ |
| L(s) = 1 | + 0.350·2-s − 0.877·4-s − 0.110·5-s + 0.512·7-s − 0.657·8-s − 0.0387·10-s − 1.33·11-s − 0.261·13-s + 0.179·14-s + 0.647·16-s + 0.316·19-s + 0.0971·20-s − 0.466·22-s − 0.431·23-s − 0.987·25-s − 0.0915·26-s − 0.450·28-s + 0.410·29-s − 1.62·31-s + 0.883·32-s − 0.0567·35-s − 0.383·37-s + 0.110·38-s + 0.0727·40-s + 1.43·41-s + 0.575·43-s + 1.16·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.126314369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.126314369\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.495T + 2T^{2} \) |
| 5 | \( 1 + 0.247T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + 4.41T + 11T^{2} \) |
| 13 | \( 1 + 0.943T + 13T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 23 | \( 1 + 2.07T + 23T^{2} \) |
| 29 | \( 1 - 2.21T + 29T^{2} \) |
| 31 | \( 1 + 9.03T + 31T^{2} \) |
| 37 | \( 1 + 2.33T + 37T^{2} \) |
| 41 | \( 1 - 9.21T + 41T^{2} \) |
| 43 | \( 1 - 3.77T + 43T^{2} \) |
| 47 | \( 1 + 1.21T + 47T^{2} \) |
| 53 | \( 1 + 8.45T + 53T^{2} \) |
| 59 | \( 1 - 6.13T + 59T^{2} \) |
| 61 | \( 1 + 0.160T + 61T^{2} \) |
| 67 | \( 1 + 0.0594T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 1.21T + 73T^{2} \) |
| 79 | \( 1 + 3.04T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 9.84T + 89T^{2} \) |
| 97 | \( 1 - 9.09T + 97T^{2} \) |
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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
−7.961237968430232014015590866754, −7.37074588391284248486214726771, −6.29639572305957232592520862661, −5.44896127156467520875457328641, −5.17666770438661976923752148106, −4.33333072762756554272648396850, −3.66903308908821223049242796118, −2.78454886616997264027449446923, −1.86257142581241532768291744477, −0.48983431595792917037558510336,
0.48983431595792917037558510336, 1.86257142581241532768291744477, 2.78454886616997264027449446923, 3.66903308908821223049242796118, 4.33333072762756554272648396850, 5.17666770438661976923752148106, 5.44896127156467520875457328641, 6.29639572305957232592520862661, 7.37074588391284248486214726771, 7.961237968430232014015590866754
