| L(s) = 1 | + 2-s + 4-s + 1.23·5-s − 2.85·7-s + 8-s + 1.23·10-s + 6.23·13-s − 2.85·14-s + 16-s − 6.23·17-s − 5·19-s + 1.23·20-s − 7.61·23-s − 3.47·25-s + 6.23·26-s − 2.85·28-s + 7.09·29-s − 7·31-s + 32-s − 6.23·34-s − 3.52·35-s + 9·37-s − 5·38-s + 1.23·40-s + 3.38·41-s + 0.708·43-s − 7.61·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.552·5-s − 1.07·7-s + 0.353·8-s + 0.390·10-s + 1.72·13-s − 0.762·14-s + 0.250·16-s − 1.51·17-s − 1.14·19-s + 0.276·20-s − 1.58·23-s − 0.694·25-s + 1.22·26-s − 0.539·28-s + 1.31·29-s − 1.25·31-s + 0.176·32-s − 1.06·34-s − 0.596·35-s + 1.47·37-s − 0.811·38-s + 0.195·40-s + 0.528·41-s + 0.108·43-s − 1.12·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 6.23T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 - 7.09T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 9T + 37T^{2} \) |
| 41 | \( 1 - 3.38T + 41T^{2} \) |
| 43 | \( 1 - 0.708T + 43T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 9.32T + 59T^{2} \) |
| 61 | \( 1 + 3.47T + 61T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 + 4.52T + 71T^{2} \) |
| 73 | \( 1 - 4.23T + 73T^{2} \) |
| 79 | \( 1 + 15T + 79T^{2} \) |
| 83 | \( 1 + 3.76T + 83T^{2} \) |
| 89 | \( 1 + 4.23T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.55352991707261344517354492790, −6.47560235194364631911237949901, −6.17849854322812970420464259835, −5.92535147972065275560616287836, −4.54958168666218842464007547393, −4.08273484857329920948505044139, −3.28243184109351546120734989951, −2.38024010030333343317825233255, −1.59268629555755643637457562019, 0,
1.59268629555755643637457562019, 2.38024010030333343317825233255, 3.28243184109351546120734989951, 4.08273484857329920948505044139, 4.54958168666218842464007547393, 5.92535147972065275560616287836, 6.17849854322812970420464259835, 6.47560235194364631911237949901, 7.55352991707261344517354492790
