| L(s) = 1 | − 1.65·2-s + 0.748·3-s + 0.755·4-s − 4.19·5-s − 1.24·6-s − 4.84·7-s + 2.06·8-s − 2.44·9-s + 6.95·10-s − 11-s + 0.565·12-s − 6.28·13-s + 8.04·14-s − 3.13·15-s − 4.94·16-s − 17-s + 4.05·18-s − 2.93·19-s − 3.16·20-s − 3.62·21-s + 1.65·22-s + 1.05·23-s + 1.54·24-s + 12.5·25-s + 10.4·26-s − 4.07·27-s − 3.65·28-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.432·3-s + 0.377·4-s − 1.87·5-s − 0.507·6-s − 1.83·7-s + 0.730·8-s − 0.813·9-s + 2.20·10-s − 0.301·11-s + 0.163·12-s − 1.74·13-s + 2.14·14-s − 0.809·15-s − 1.23·16-s − 0.242·17-s + 0.954·18-s − 0.672·19-s − 0.707·20-s − 0.790·21-s + 0.353·22-s + 0.219·23-s + 0.315·24-s + 2.51·25-s + 2.04·26-s − 0.783·27-s − 0.691·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 3 | \( 1 - 0.748T + 3T^{2} \) |
| 5 | \( 1 + 4.19T + 5T^{2} \) |
| 7 | \( 1 + 4.84T + 7T^{2} \) |
| 13 | \( 1 + 6.28T + 13T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 + 3.95T + 29T^{2} \) |
| 31 | \( 1 - 8.30T + 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 47 | \( 1 - 0.558T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 - 15.2T + 59T^{2} \) |
| 61 | \( 1 + 5.01T + 61T^{2} \) |
| 67 | \( 1 - 1.59T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 6.68T + 73T^{2} \) |
| 79 | \( 1 + 0.622T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 + 9.83T + 89T^{2} \) |
| 97 | \( 1 + 6.90T + 97T^{2} \) |
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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
−7.45068487275710253355771916238, −7.22707915031745877289578615611, −6.54921930512005432410905447641, −5.35651655704514572223077105438, −4.43679077331383586643033438306, −3.79308168396772767146636335237, −2.96181329938245598878087329234, −2.41913450830850318632166436065, −0.52725937509815735467575552514, 0,
0.52725937509815735467575552514, 2.41913450830850318632166436065, 2.96181329938245598878087329234, 3.79308168396772767146636335237, 4.43679077331383586643033438306, 5.35651655704514572223077105438, 6.54921930512005432410905447641, 7.22707915031745877289578615611, 7.45068487275710253355771916238
