| L(s) = 1 | − 2-s + 3-s + 4-s + 0.876·5-s − 6-s − 4.57·7-s − 8-s + 9-s − 0.876·10-s + 3.41·11-s + 12-s − 5.45·13-s + 4.57·14-s + 0.876·15-s + 16-s + 17-s − 18-s − 0.613·19-s + 0.876·20-s − 4.57·21-s − 3.41·22-s + 8.17·23-s − 24-s − 4.23·25-s + 5.45·26-s + 27-s − 4.57·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.392·5-s − 0.408·6-s − 1.73·7-s − 0.353·8-s + 0.333·9-s − 0.277·10-s + 1.02·11-s + 0.288·12-s − 1.51·13-s + 1.22·14-s + 0.226·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 0.140·19-s + 0.196·20-s − 0.999·21-s − 0.728·22-s + 1.70·23-s − 0.204·24-s − 0.846·25-s + 1.07·26-s + 0.192·27-s − 0.865·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 0.876T + 5T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 19 | \( 1 + 0.613T + 19T^{2} \) |
| 23 | \( 1 - 8.17T + 23T^{2} \) |
| 29 | \( 1 - 0.557T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 + 4.45T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 - 0.576T + 43T^{2} \) |
| 47 | \( 1 - 2.49T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 0.345T + 67T^{2} \) |
| 71 | \( 1 - 5.75T + 71T^{2} \) |
| 73 | \( 1 + 6.22T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 5.00T + 83T^{2} \) |
| 89 | \( 1 - 8.09T + 89T^{2} \) |
| 97 | \( 1 - 6.31T + 97T^{2} \) |
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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
−7.60501243912791663890666691933, −7.06999589251063212982334004080, −6.54459100871018319315380942240, −5.85059582884057577723016129299, −4.80990561150290440875420656527, −3.77623361938670888490979853475, −3.00619557700050044867619478449, −2.44136413100086568310123709684, −1.26708216073879538998614855818, 0,
1.26708216073879538998614855818, 2.44136413100086568310123709684, 3.00619557700050044867619478449, 3.77623361938670888490979853475, 4.80990561150290440875420656527, 5.85059582884057577723016129299, 6.54459100871018319315380942240, 7.06999589251063212982334004080, 7.60501243912791663890666691933
