| L(s) = 1 | + 2.23·2-s − 2.61·3-s + 3.00·4-s − 5.85·6-s + 3.23·7-s + 2.23·8-s + 3.85·9-s + 0.618·11-s − 7.85·12-s + 3.85·13-s + 7.23·14-s − 0.999·16-s − 6.09·17-s + 8.61·18-s − 7.61·19-s − 8.47·21-s + 1.38·22-s − 4.47·23-s − 5.85·24-s + 8.61·26-s − 2.23·27-s + 9.70·28-s + 4.09·29-s − 6.47·31-s − 6.70·32-s − 1.61·33-s − 13.6·34-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 1.51·3-s + 1.50·4-s − 2.38·6-s + 1.22·7-s + 0.790·8-s + 1.28·9-s + 0.186·11-s − 2.26·12-s + 1.06·13-s + 1.93·14-s − 0.249·16-s − 1.47·17-s + 2.03·18-s − 1.74·19-s − 1.84·21-s + 0.294·22-s − 0.932·23-s − 1.19·24-s + 1.69·26-s − 0.430·27-s + 1.83·28-s + 0.759·29-s − 1.16·31-s − 1.18·32-s − 0.281·33-s − 2.33·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 - 0.618T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 + 6.09T + 17T^{2} \) |
| 19 | \( 1 + 7.61T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 4.61T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 5.61T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 + 5.70T + 59T^{2} \) |
| 61 | \( 1 - 6.14T + 61T^{2} \) |
| 67 | \( 1 - 4.09T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 7.61T + 73T^{2} \) |
| 79 | \( 1 + 6.18T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 + 2.76T + 97T^{2} \) |
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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
−7.26910898099974260737272675032, −6.49416157457007506918104019709, −6.19341451408584374871711381940, −5.49598776227145022730825417455, −4.77723661938273495426333475556, −4.34780276917949215552720148122, −3.74166410150022803688847032439, −2.29595540104545699431139219537, −1.57525249054104748864286375215, 0,
1.57525249054104748864286375215, 2.29595540104545699431139219537, 3.74166410150022803688847032439, 4.34780276917949215552720148122, 4.77723661938273495426333475556, 5.49598776227145022730825417455, 6.19341451408584374871711381940, 6.49416157457007506918104019709, 7.26910898099974260737272675032
