| L(s) = 1 | − 2.18·2-s + 2.15·3-s + 2.77·4-s − 3.43·5-s − 4.71·6-s + 3.93·7-s − 1.69·8-s + 1.65·9-s + 7.49·10-s + 11-s + 5.98·12-s + 3.31·13-s − 8.60·14-s − 7.40·15-s − 1.84·16-s + 2.80·17-s − 3.61·18-s − 19-s − 9.52·20-s + 8.50·21-s − 2.18·22-s + 6.88·23-s − 3.65·24-s + 6.77·25-s − 7.23·26-s − 2.90·27-s + 10.9·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.24·3-s + 1.38·4-s − 1.53·5-s − 1.92·6-s + 1.48·7-s − 0.598·8-s + 0.551·9-s + 2.37·10-s + 0.301·11-s + 1.72·12-s + 0.918·13-s − 2.30·14-s − 1.91·15-s − 0.462·16-s + 0.680·17-s − 0.852·18-s − 0.229·19-s − 2.12·20-s + 1.85·21-s − 0.465·22-s + 1.43·23-s − 0.746·24-s + 1.35·25-s − 1.41·26-s − 0.558·27-s + 2.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8128565125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8128565125\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 3 | \( 1 - 2.15T + 3T^{2} \) |
| 5 | \( 1 + 3.43T + 5T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 13 | \( 1 - 3.31T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 23 | \( 1 - 6.88T + 23T^{2} \) |
| 29 | \( 1 - 5.67T + 29T^{2} \) |
| 31 | \( 1 - 2.51T + 31T^{2} \) |
| 37 | \( 1 + 6.39T + 37T^{2} \) |
| 41 | \( 1 - 0.560T + 41T^{2} \) |
| 43 | \( 1 + 9.40T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 5.68T + 53T^{2} \) |
| 59 | \( 1 - 4.35T + 59T^{2} \) |
| 61 | \( 1 + 3.56T + 61T^{2} \) |
| 67 | \( 1 + 9.95T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 8.95T + 73T^{2} \) |
| 79 | \( 1 - 8.49T + 79T^{2} \) |
| 83 | \( 1 + 5.21T + 83T^{2} \) |
| 89 | \( 1 + 7.28T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.82243334920565993602260658694, −11.29684397023825067564014669926, −10.32443070872693083708570317500, −8.856293721846946901344435522648, −8.434794875586729689328012238226, −7.913278551149302873129432667658, −7.04481354140097962126135527504, −4.60654322182126309154676167960, −3.24052470619799226856810859236, −1.41212230524311918316921922786,
1.41212230524311918316921922786, 3.24052470619799226856810859236, 4.60654322182126309154676167960, 7.04481354140097962126135527504, 7.913278551149302873129432667658, 8.434794875586729689328012238226, 8.856293721846946901344435522648, 10.32443070872693083708570317500, 11.29684397023825067564014669926, 11.82243334920565993602260658694
