L(s) = 1 | + 1.91·2-s + 0.186·3-s + 1.65·4-s + 0.355·6-s − 3.52·7-s − 0.663·8-s − 2.96·9-s + 0.515·11-s + 0.307·12-s + 5.38·13-s − 6.74·14-s − 4.57·16-s + 4.16·17-s − 5.66·18-s + 4.92·19-s − 0.657·21-s + 0.985·22-s + 7.69·23-s − 0.123·24-s + 10.2·26-s − 1.11·27-s − 5.83·28-s − 8.93·29-s − 4.43·31-s − 7.41·32-s + 0.0959·33-s + 7.96·34-s + ⋯ |
L(s) = 1 | + 1.35·2-s + 0.107·3-s + 0.826·4-s + 0.145·6-s − 1.33·7-s − 0.234·8-s − 0.988·9-s + 0.155·11-s + 0.0888·12-s + 1.49·13-s − 1.80·14-s − 1.14·16-s + 1.01·17-s − 1.33·18-s + 1.13·19-s − 0.143·21-s + 0.210·22-s + 1.60·23-s − 0.0252·24-s + 2.01·26-s − 0.213·27-s − 1.10·28-s − 1.65·29-s − 0.795·31-s − 1.31·32-s + 0.0167·33-s + 1.36·34-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 | \( 1 - 1.91T + 2T^{2} \) |
| 3 | \( 1 - 0.186T + 3T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 - 0.515T + 11T^{2} \) |
| 13 | \( 1 - 5.38T + 13T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 19 | \( 1 - 4.92T + 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 + 5.99T + 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 - 1.66T + 43T^{2} \) |
| 47 | \( 1 + 8.55T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 9.25T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 3.91T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 1.43T + 79T^{2} \) |
| 83 | \( 1 + 1.73T + 83T^{2} \) |
| 89 | \( 1 + 1.07T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.45781382945161607937413221660, −6.74558119704038801557956336352, −6.04742598004296656365346102303, −5.58453954017203287209056791790, −4.99125423344901208041356618688, −3.71447690082745378758782359501, −3.27782049858611290373735521444, −3.05791456243855914230237755041, −1.51514474603164630693546597119, 0,
1.51514474603164630693546597119, 3.05791456243855914230237755041, 3.27782049858611290373735521444, 3.71447690082745378758782359501, 4.99125423344901208041356618688, 5.58453954017203287209056791790, 6.04742598004296656365346102303, 6.74558119704038801557956336352, 7.45781382945161607937413221660