| L(s) = 1 | + 2.25·2-s + 3-s + 3.10·4-s + 2.25·6-s − 3.65·7-s + 2.48·8-s + 9-s − 5.39·11-s + 3.10·12-s − 3.76·13-s − 8.24·14-s − 0.583·16-s + 6.40·17-s + 2.25·18-s − 8.07·19-s − 3.65·21-s − 12.1·22-s + 4.76·23-s + 2.48·24-s − 8.50·26-s + 27-s − 11.3·28-s − 3.78·29-s − 5.26·31-s − 6.29·32-s − 5.39·33-s + 14.4·34-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.577·3-s + 1.55·4-s + 0.922·6-s − 1.38·7-s + 0.879·8-s + 0.333·9-s − 1.62·11-s + 0.895·12-s − 1.04·13-s − 2.20·14-s − 0.145·16-s + 1.55·17-s + 0.532·18-s − 1.85·19-s − 0.797·21-s − 2.59·22-s + 0.993·23-s + 0.507·24-s − 1.66·26-s + 0.192·27-s − 2.14·28-s − 0.702·29-s − 0.944·31-s − 1.11·32-s − 0.938·33-s + 2.48·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 2.25T + 2T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 + 5.39T + 11T^{2} \) |
| 13 | \( 1 + 3.76T + 13T^{2} \) |
| 17 | \( 1 - 6.40T + 17T^{2} \) |
| 19 | \( 1 + 8.07T + 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 + 3.78T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 + 3.48T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 53 | \( 1 - 9.62T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 4.68T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 1.64T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 7.50T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−7.962055732268475466275516238061, −7.20902068478167681997804187826, −6.65067405199159768794126680655, −5.61861655319487134264166556109, −5.26043909365166899737960747154, −4.23876889424857307017463114391, −3.44223257485394838157142732881, −2.81798701677671439179171381901, −2.20896956125419026396511463892, 0,
2.20896956125419026396511463892, 2.81798701677671439179171381901, 3.44223257485394838157142732881, 4.23876889424857307017463114391, 5.26043909365166899737960747154, 5.61861655319487134264166556109, 6.65067405199159768794126680655, 7.20902068478167681997804187826, 7.962055732268475466275516238061
