| L(s) = 1 | + 2.24·2-s − 0.790·3-s + 3.06·4-s − 3.96·5-s − 1.77·6-s − 4.97·7-s + 2.38·8-s − 2.37·9-s − 8.93·10-s + 6.10·11-s − 2.42·12-s − 0.944·13-s − 11.1·14-s + 3.13·15-s − 0.753·16-s + 0.884·17-s − 5.34·18-s − 2.59·19-s − 12.1·20-s + 3.93·21-s + 13.7·22-s − 2.77·23-s − 1.88·24-s + 10.7·25-s − 2.12·26-s + 4.25·27-s − 15.2·28-s + ⋯ |
| L(s) = 1 | + 1.59·2-s − 0.456·3-s + 1.53·4-s − 1.77·5-s − 0.726·6-s − 1.88·7-s + 0.843·8-s − 0.791·9-s − 2.82·10-s + 1.84·11-s − 0.698·12-s − 0.262·13-s − 2.99·14-s + 0.810·15-s − 0.188·16-s + 0.214·17-s − 1.25·18-s − 0.596·19-s − 2.71·20-s + 0.858·21-s + 2.92·22-s − 0.577·23-s − 0.385·24-s + 2.15·25-s − 0.416·26-s + 0.818·27-s − 2.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 547 | \( 1 + T \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 3 | \( 1 + 0.790T + 3T^{2} \) |
| 5 | \( 1 + 3.96T + 5T^{2} \) |
| 7 | \( 1 + 4.97T + 7T^{2} \) |
| 11 | \( 1 - 6.10T + 11T^{2} \) |
| 13 | \( 1 + 0.944T + 13T^{2} \) |
| 17 | \( 1 - 0.884T + 17T^{2} \) |
| 19 | \( 1 + 2.59T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 + 1.93T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 - 1.37T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 2.84T + 47T^{2} \) |
| 53 | \( 1 - 6.58T + 53T^{2} \) |
| 59 | \( 1 - 0.838T + 59T^{2} \) |
| 61 | \( 1 + 9.71T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 7.51T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 0.398T + 79T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 3.97T + 97T^{2} \) |
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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
−10.89602014285924081079803558867, −9.499613370313582155795982576693, −8.534781314766289633945610235685, −7.05015923542252856252033877159, −6.56051400324832413215570162924, −5.73488530902990453408523817742, −4.31967532183064974295782030991, −3.71119906215551841395019411209, −3.03824215480465189659466127013, 0,
3.03824215480465189659466127013, 3.71119906215551841395019411209, 4.31967532183064974295782030991, 5.73488530902990453408523817742, 6.56051400324832413215570162924, 7.05015923542252856252033877159, 8.534781314766289633945610235685, 9.499613370313582155795982576693, 10.89602014285924081079803558867
