| L(s) = 1 | + 1.80·2-s + 3-s + 1.24·4-s − 5-s + 1.80·6-s − 4·7-s − 1.35·8-s + 9-s − 1.80·10-s + 4.09·11-s + 1.24·12-s − 7.20·14-s − 15-s − 4.93·16-s + 6.69·17-s + 1.80·18-s + 3.82·19-s − 1.24·20-s − 4·21-s + 7.38·22-s + 6.45·23-s − 1.35·24-s + 25-s + 27-s − 4.98·28-s + 0.219·29-s − 1.80·30-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.577·3-s + 0.623·4-s − 0.447·5-s + 0.735·6-s − 1.51·7-s − 0.479·8-s + 0.333·9-s − 0.569·10-s + 1.23·11-s + 0.359·12-s − 1.92·14-s − 0.258·15-s − 1.23·16-s + 1.62·17-s + 0.424·18-s + 0.878·19-s − 0.278·20-s − 0.872·21-s + 1.57·22-s + 1.34·23-s − 0.276·24-s + 0.200·25-s + 0.192·27-s − 0.942·28-s + 0.0408·29-s − 0.328·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.598903088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.598903088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 4.09T + 11T^{2} \) |
| 17 | \( 1 - 6.69T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 - 6.45T + 23T^{2} \) |
| 29 | \( 1 - 0.219T + 29T^{2} \) |
| 31 | \( 1 - 8.51T + 31T^{2} \) |
| 37 | \( 1 - 2.93T + 37T^{2} \) |
| 41 | \( 1 + 7.48T + 41T^{2} \) |
| 43 | \( 1 - 6.31T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 0.225T + 53T^{2} \) |
| 59 | \( 1 - 1.87T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 8.05T + 67T^{2} \) |
| 71 | \( 1 + 4.98T + 71T^{2} \) |
| 73 | \( 1 + 4.49T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 4.81T + 89T^{2} \) |
| 97 | \( 1 - 3.50T + 97T^{2} \) |
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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
−9.025903780991276391301486445302, −8.106816413060229021530747198036, −7.01647430069470935930184704539, −6.59044657106165504466102828933, −5.73536203255904041910898247025, −4.83721423751947877046345422146, −3.86463659763401368428484774082, −3.30724868783284903130372209300, −2.85703543651760276957597676888, −1.00726169374457950233428562019,
1.00726169374457950233428562019, 2.85703543651760276957597676888, 3.30724868783284903130372209300, 3.86463659763401368428484774082, 4.83721423751947877046345422146, 5.73536203255904041910898247025, 6.59044657106165504466102828933, 7.01647430069470935930184704539, 8.106816413060229021530747198036, 9.025903780991276391301486445302
