L(s) = 1 | − 2-s + 3.28·3-s + 4-s + 1.08·5-s − 3.28·6-s + 0.110·7-s − 8-s + 7.77·9-s − 1.08·10-s − 0.681·11-s + 3.28·12-s + 2.71·13-s − 0.110·14-s + 3.55·15-s + 16-s + 1.35·17-s − 7.77·18-s + 6.28·19-s + 1.08·20-s + 0.363·21-s + 0.681·22-s − 4.80·23-s − 3.28·24-s − 3.83·25-s − 2.71·26-s + 15.6·27-s + 0.110·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.89·3-s + 0.5·4-s + 0.483·5-s − 1.34·6-s + 0.0418·7-s − 0.353·8-s + 2.59·9-s − 0.342·10-s − 0.205·11-s + 0.947·12-s + 0.753·13-s − 0.0295·14-s + 0.916·15-s + 0.250·16-s + 0.329·17-s − 1.83·18-s + 1.44·19-s + 0.241·20-s + 0.0792·21-s + 0.145·22-s − 1.00·23-s − 0.670·24-s − 0.766·25-s − 0.533·26-s + 3.01·27-s + 0.0209·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.506778214\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.506778214\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 - 3.28T + 3T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 7 | \( 1 - 0.110T + 7T^{2} \) |
| 11 | \( 1 + 0.681T + 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 - 6.28T + 19T^{2} \) |
| 23 | \( 1 + 4.80T + 23T^{2} \) |
| 29 | \( 1 + 0.955T + 29T^{2} \) |
| 31 | \( 1 - 0.910T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 2.86T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 + 0.689T + 53T^{2} \) |
| 59 | \( 1 + 4.59T + 59T^{2} \) |
| 61 | \( 1 + 8.76T + 61T^{2} \) |
| 67 | \( 1 - 7.64T + 67T^{2} \) |
| 71 | \( 1 - 5.40T + 71T^{2} \) |
| 73 | \( 1 - 8.03T + 73T^{2} \) |
| 79 | \( 1 + 0.472T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 5.86T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.253496409861553239979096110714, −8.018002375312116907954546674279, −7.38662630890021420737374305453, −6.47265779301668017839405088335, −5.61195724046458826039748083033, −4.37539028787411711404654141390, −3.50573583411480625192478633852, −2.85272378611762853805335808205, −1.97483612125103865108365221188, −1.21209221992483881164425675282,
1.21209221992483881164425675282, 1.97483612125103865108365221188, 2.85272378611762853805335808205, 3.50573583411480625192478633852, 4.37539028787411711404654141390, 5.61195724046458826039748083033, 6.47265779301668017839405088335, 7.38662630890021420737374305453, 8.018002375312116907954546674279, 8.253496409861553239979096110714