| L(s) = 1 | − 0.0567·2-s − 2.90·3-s − 7.99·4-s + 7.08·5-s + 0.164·6-s − 24.8·7-s + 0.907·8-s − 18.5·9-s − 0.401·10-s + 23.2·12-s + 20.4·13-s + 1.40·14-s − 20.5·15-s + 63.9·16-s − 17·17-s + 1.05·18-s + 86.2·19-s − 56.6·20-s + 72.1·21-s − 201.·23-s − 2.63·24-s − 74.8·25-s − 1.15·26-s + 132.·27-s + 198.·28-s + 203.·29-s + 1.16·30-s + ⋯ |
| L(s) = 1 | − 0.0200·2-s − 0.559·3-s − 0.999·4-s + 0.633·5-s + 0.0112·6-s − 1.34·7-s + 0.0401·8-s − 0.686·9-s − 0.0127·10-s + 0.559·12-s + 0.435·13-s + 0.0268·14-s − 0.354·15-s + 0.998·16-s − 0.242·17-s + 0.0137·18-s + 1.04·19-s − 0.633·20-s + 0.750·21-s − 1.82·23-s − 0.0224·24-s − 0.598·25-s − 0.00873·26-s + 0.944·27-s + 1.33·28-s + 1.30·29-s + 0.00711·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 0.0567T + 8T^{2} \) |
| 3 | \( 1 + 2.90T + 27T^{2} \) |
| 5 | \( 1 - 7.08T + 125T^{2} \) |
| 7 | \( 1 + 24.8T + 343T^{2} \) |
| 13 | \( 1 - 20.4T + 2.19e3T^{2} \) |
| 19 | \( 1 - 86.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 201.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 203.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 215.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 230.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 192.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 110.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 192.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 373.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 127.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 587.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 81.6T + 3.00e5T^{2} \) |
| 71 | \( 1 - 204.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 735.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 486.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 178.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 768.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 957.T + 9.12e5T^{2} \) |
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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
−8.561173469829537825987883016887, −7.66837464960244320707751786445, −6.52183051513622090816673811332, −5.83056347716609994851184048311, −5.50133415826989702251039905876, −4.25242851453273533367298747255, −3.48106512630670574588635861858, −2.44824437858969688188318135998, −0.891360829208873224842462390094, 0,
0.891360829208873224842462390094, 2.44824437858969688188318135998, 3.48106512630670574588635861858, 4.25242851453273533367298747255, 5.50133415826989702251039905876, 5.83056347716609994851184048311, 6.52183051513622090816673811332, 7.66837464960244320707751786445, 8.561173469829537825987883016887
