L(s) = 1 | + 1.75·2-s + 0.0856·3-s + 1.08·4-s + 5-s + 0.150·6-s − 1.86·7-s − 1.60·8-s − 2.99·9-s + 1.75·10-s − 1.22·11-s + 0.0929·12-s − 2.40·13-s − 3.26·14-s + 0.0856·15-s − 4.99·16-s + 6.05·17-s − 5.25·18-s + 1.08·20-s − 0.159·21-s − 2.15·22-s − 7.20·23-s − 0.137·24-s + 25-s − 4.22·26-s − 0.513·27-s − 2.02·28-s + 1.28·29-s + ⋯ |
L(s) = 1 | + 1.24·2-s + 0.0494·3-s + 0.542·4-s + 0.447·5-s + 0.0614·6-s − 0.703·7-s − 0.567·8-s − 0.997·9-s + 0.555·10-s − 0.369·11-s + 0.0268·12-s − 0.667·13-s − 0.873·14-s + 0.0221·15-s − 1.24·16-s + 1.46·17-s − 1.23·18-s + 0.242·20-s − 0.0347·21-s − 0.458·22-s − 1.50·23-s − 0.0280·24-s + 0.200·25-s − 0.829·26-s − 0.0987·27-s − 0.381·28-s + 0.238·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 1.75T + 2T^{2} \) |
| 3 | \( 1 - 0.0856T + 3T^{2} \) |
| 7 | \( 1 + 1.86T + 7T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 + 2.40T + 13T^{2} \) |
| 17 | \( 1 - 6.05T + 17T^{2} \) |
| 23 | \( 1 + 7.20T + 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 + 5.87T + 31T^{2} \) |
| 37 | \( 1 + 1.59T + 37T^{2} \) |
| 41 | \( 1 + 9.72T + 41T^{2} \) |
| 43 | \( 1 + 0.697T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 9.40T + 59T^{2} \) |
| 61 | \( 1 - 3.73T + 61T^{2} \) |
| 67 | \( 1 - 7.41T + 67T^{2} \) |
| 71 | \( 1 + 8.51T + 71T^{2} \) |
| 73 | \( 1 + 6.21T + 73T^{2} \) |
| 79 | \( 1 + 7.39T + 79T^{2} \) |
| 83 | \( 1 - 0.0135T + 83T^{2} \) |
| 89 | \( 1 + 1.25T + 89T^{2} \) |
| 97 | \( 1 - 7.97T + 97T^{2} \) |
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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
−8.871831568507156595954593274596, −8.087420512136169007297023690388, −7.02240182901016060334251025893, −6.12031500498692171299919701468, −5.55493649976042875383061307124, −4.96298457855662896642007588427, −3.67953200170138527747880619613, −3.12967804780164720681069809144, −2.14910618556348512450075785163, 0,
2.14910618556348512450075785163, 3.12967804780164720681069809144, 3.67953200170138527747880619613, 4.96298457855662896642007588427, 5.55493649976042875383061307124, 6.12031500498692171299919701468, 7.02240182901016060334251025893, 8.087420512136169007297023690388, 8.871831568507156595954593274596