| L(s) = 1 | − 1.45·2-s − 2.70·3-s + 0.106·4-s + 0.925·5-s + 3.91·6-s + 0.794·7-s + 2.74·8-s + 4.29·9-s − 1.34·10-s + 2.13·11-s − 0.287·12-s + 0.718·13-s − 1.15·14-s − 2.49·15-s − 4.20·16-s − 1.47·17-s − 6.23·18-s + 0.352·19-s + 0.0986·20-s − 2.14·21-s − 3.10·22-s − 4.41·23-s − 7.42·24-s − 4.14·25-s − 1.04·26-s − 3.49·27-s + 0.0846·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s − 1.55·3-s + 0.0532·4-s + 0.413·5-s + 1.60·6-s + 0.300·7-s + 0.971·8-s + 1.43·9-s − 0.424·10-s + 0.644·11-s − 0.0830·12-s + 0.199·13-s − 0.308·14-s − 0.645·15-s − 1.05·16-s − 0.358·17-s − 1.46·18-s + 0.0809·19-s + 0.0220·20-s − 0.468·21-s − 0.661·22-s − 0.919·23-s − 1.51·24-s − 0.828·25-s − 0.204·26-s − 0.672·27-s + 0.0160·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 3 | \( 1 + 2.70T + 3T^{2} \) |
| 5 | \( 1 - 0.925T + 5T^{2} \) |
| 7 | \( 1 - 0.794T + 7T^{2} \) |
| 11 | \( 1 - 2.13T + 11T^{2} \) |
| 13 | \( 1 - 0.718T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 - 0.352T + 19T^{2} \) |
| 23 | \( 1 + 4.41T + 23T^{2} \) |
| 29 | \( 1 + 6.22T + 29T^{2} \) |
| 31 | \( 1 - 0.577T + 31T^{2} \) |
| 37 | \( 1 + 3.85T + 37T^{2} \) |
| 41 | \( 1 - 9.56T + 41T^{2} \) |
| 43 | \( 1 - 2.17T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 9.19T + 59T^{2} \) |
| 61 | \( 1 + 1.10T + 61T^{2} \) |
| 67 | \( 1 - 4.36T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 + 3.79T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 7.64T + 83T^{2} \) |
| 89 | \( 1 - 3.69T + 89T^{2} \) |
| 97 | \( 1 + 0.902T + 97T^{2} \) |
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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
−8.616740678517328516934673170244, −7.64024866093640840141779794968, −7.04245050563996053794057735465, −6.03660657952528116394750191711, −5.63801263432534737325733688305, −4.58421593714324028042497318724, −3.96347974176311081263643690342, −2.04249087301766508356189951320, −1.13663377253065788523916713971, 0,
1.13663377253065788523916713971, 2.04249087301766508356189951320, 3.96347974176311081263643690342, 4.58421593714324028042497318724, 5.63801263432534737325733688305, 6.03660657952528116394750191711, 7.04245050563996053794057735465, 7.64024866093640840141779794968, 8.616740678517328516934673170244
