| L(s) = 1 | + 1.72·3-s + 3.13·5-s + 7-s − 0.0378·9-s − 4.35·11-s − 13-s + 5.39·15-s + 5.31·17-s + 2.49·19-s + 1.72·21-s + 2.85·23-s + 4.81·25-s − 5.22·27-s + 3.62·29-s + 6.17·31-s − 7.49·33-s + 3.13·35-s + 9.79·37-s − 1.72·39-s + 10.6·41-s + 0.449·43-s − 0.118·45-s − 11.8·47-s + 49-s + 9.15·51-s + 5.89·53-s − 13.6·55-s + ⋯ |
| L(s) = 1 | + 0.993·3-s + 1.40·5-s + 0.377·7-s − 0.0126·9-s − 1.31·11-s − 0.277·13-s + 1.39·15-s + 1.28·17-s + 0.573·19-s + 0.375·21-s + 0.595·23-s + 0.963·25-s − 1.00·27-s + 0.673·29-s + 1.10·31-s − 1.30·33-s + 0.529·35-s + 1.61·37-s − 0.275·39-s + 1.66·41-s + 0.0686·43-s − 0.0176·45-s − 1.73·47-s + 0.142·49-s + 1.28·51-s + 0.809·53-s − 1.83·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.015004912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.015004912\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 - 3.13T + 5T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 - 3.62T + 29T^{2} \) |
| 31 | \( 1 - 6.17T + 31T^{2} \) |
| 37 | \( 1 - 9.79T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 0.449T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 5.89T + 53T^{2} \) |
| 59 | \( 1 - 7.80T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + 5.84T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 4.17T + 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 5.57T + 89T^{2} \) |
| 97 | \( 1 + 0.979T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.703217359900474251445418825829, −8.701907357748347169391643993256, −7.972427093838319773657585954798, −7.36987299301824531663029701179, −6.00865630296110725548163227901, −5.49606750649498744656827555361, −4.53967379799196774544207541730, −2.88691224930538427592004375095, −2.67697632614240162081736555124, −1.34811907969582490950115101205,
1.34811907969582490950115101205, 2.67697632614240162081736555124, 2.88691224930538427592004375095, 4.53967379799196774544207541730, 5.49606750649498744656827555361, 6.00865630296110725548163227901, 7.36987299301824531663029701179, 7.972427093838319773657585954798, 8.701907357748347169391643993256, 9.703217359900474251445418825829
