| L(s) = 1 | − 2.55·2-s − 0.410·3-s + 4.51·4-s + 3.42·5-s + 1.04·6-s − 7-s − 6.41·8-s − 2.83·9-s − 8.75·10-s − 1.85·12-s + 2.17·13-s + 2.55·14-s − 1.40·15-s + 7.33·16-s + 4.51·17-s + 7.22·18-s + 2.42·19-s + 15.4·20-s + 0.410·21-s + 0.648·23-s + 2.63·24-s + 6.75·25-s − 5.55·26-s + 2.39·27-s − 4.51·28-s + 1.25·29-s + 3.59·30-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.237·3-s + 2.25·4-s + 1.53·5-s + 0.427·6-s − 0.377·7-s − 2.26·8-s − 0.943·9-s − 2.76·10-s − 0.534·12-s + 0.603·13-s + 0.682·14-s − 0.363·15-s + 1.83·16-s + 1.09·17-s + 1.70·18-s + 0.556·19-s + 3.45·20-s + 0.0895·21-s + 0.135·23-s + 0.537·24-s + 1.35·25-s − 1.08·26-s + 0.460·27-s − 0.852·28-s + 0.232·29-s + 0.655·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7765459947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7765459947\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 + 0.410T + 3T^{2} \) |
| 5 | \( 1 - 3.42T + 5T^{2} \) |
| 13 | \( 1 - 2.17T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 - 0.648T + 23T^{2} \) |
| 29 | \( 1 - 1.25T + 29T^{2} \) |
| 31 | \( 1 + 8.03T + 31T^{2} \) |
| 37 | \( 1 - 5.01T + 37T^{2} \) |
| 41 | \( 1 - 2.62T + 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 + 5.04T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 7.66T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 6.22T + 67T^{2} \) |
| 71 | \( 1 - 4.22T + 71T^{2} \) |
| 73 | \( 1 - 5.92T + 73T^{2} \) |
| 79 | \( 1 + 9.76T + 79T^{2} \) |
| 83 | \( 1 - 8.37T + 83T^{2} \) |
| 89 | \( 1 - 4.76T + 89T^{2} \) |
| 97 | \( 1 + 8.70T + 97T^{2} \) |
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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
−9.974838462685784914035287503513, −9.381483816918126438790910246585, −8.787907471660193169793261996024, −7.85939129750210833450272232153, −6.85731424198988736219984162355, −5.98292401433093843115837976650, −5.49308941684741836306838742221, −3.18039130559851518157137346286, −2.14935009892522679818464239262, −0.966097361046974103946726300778,
0.966097361046974103946726300778, 2.14935009892522679818464239262, 3.18039130559851518157137346286, 5.49308941684741836306838742221, 5.98292401433093843115837976650, 6.85731424198988736219984162355, 7.85939129750210833450272232153, 8.787907471660193169793261996024, 9.381483816918126438790910246585, 9.974838462685784914035287503513
