L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s + 2.41·6-s − 7-s − 4.41·8-s + 9-s − 2·11-s − 3.82·12-s + 2.41·14-s + 2.99·16-s − 0.828·17-s − 2.41·18-s + 5.65·19-s + 21-s + 4.82·22-s + 1.17·23-s + 4.41·24-s − 5·25-s − 27-s − 3.82·28-s − 3.65·29-s − 1.65·31-s + 1.58·32-s + 2·33-s + 1.99·34-s + 3.82·36-s + ⋯ |
L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s + 0.985·6-s − 0.377·7-s − 1.56·8-s + 0.333·9-s − 0.603·11-s − 1.10·12-s + 0.645·14-s + 0.749·16-s − 0.200·17-s − 0.569·18-s + 1.29·19-s + 0.218·21-s + 1.02·22-s + 0.244·23-s + 0.901·24-s − 25-s − 0.192·27-s − 0.723·28-s − 0.679·29-s − 0.297·31-s + 0.280·32-s + 0.348·33-s + 0.342·34-s + 0.638·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 - 7.65T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.264327237365188141958637622305, −7.36111289784737115780983659547, −7.21101332027950977120221985613, −6.06435023580241889068945571360, −5.53632582481203793890680934465, −4.35382642449064262643723392224, −3.13377931740056975977858363869, −2.15057719051270198870369330596, −1.06181339963306821650290580281, 0,
1.06181339963306821650290580281, 2.15057719051270198870369330596, 3.13377931740056975977858363869, 4.35382642449064262643723392224, 5.53632582481203793890680934465, 6.06435023580241889068945571360, 7.21101332027950977120221985613, 7.36111289784737115780983659547, 8.264327237365188141958637622305