| L(s) = 1 | − 2.55·2-s − 3-s + 4.50·4-s − 5-s + 2.55·6-s − 6.39·8-s + 9-s + 2.55·10-s − 11-s − 4.50·12-s + 2.86·13-s + 15-s + 7.29·16-s − 4.75·17-s − 2.55·18-s + 8.17·19-s − 4.50·20-s + 2.55·22-s − 6.86·23-s + 6.39·24-s + 25-s − 7.31·26-s − 27-s + 0.937·29-s − 2.55·30-s − 3.80·31-s − 5.81·32-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.577·3-s + 2.25·4-s − 0.447·5-s + 1.04·6-s − 2.25·8-s + 0.333·9-s + 0.806·10-s − 0.301·11-s − 1.30·12-s + 0.795·13-s + 0.258·15-s + 1.82·16-s − 1.15·17-s − 0.601·18-s + 1.87·19-s − 1.00·20-s + 0.543·22-s − 1.43·23-s + 1.30·24-s + 0.200·25-s − 1.43·26-s − 0.192·27-s + 0.174·29-s − 0.465·30-s − 0.684·31-s − 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 13 | \( 1 - 2.86T + 13T^{2} \) |
| 17 | \( 1 + 4.75T + 17T^{2} \) |
| 19 | \( 1 - 8.17T + 19T^{2} \) |
| 23 | \( 1 + 6.86T + 23T^{2} \) |
| 29 | \( 1 - 0.937T + 29T^{2} \) |
| 31 | \( 1 + 3.80T + 31T^{2} \) |
| 37 | \( 1 - 4.79T + 37T^{2} \) |
| 41 | \( 1 - 5.39T + 41T^{2} \) |
| 43 | \( 1 - 0.363T + 43T^{2} \) |
| 47 | \( 1 + 4.88T + 47T^{2} \) |
| 53 | \( 1 - 2.22T + 53T^{2} \) |
| 59 | \( 1 + 2.52T + 59T^{2} \) |
| 61 | \( 1 + 2.99T + 61T^{2} \) |
| 67 | \( 1 + 5.93T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 3.99T + 79T^{2} \) |
| 83 | \( 1 - 8.14T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
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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
−7.54842092800110084942822443733, −7.13949292316811578804044607765, −6.25027473310126323484408443265, −5.79594203472025299800211051449, −4.71515772321170202464110419962, −3.75700289790538821405026449260, −2.78786243989437657875232494230, −1.81732428261924996790995353276, −0.947416142175174395082558116263, 0,
0.947416142175174395082558116263, 1.81732428261924996790995353276, 2.78786243989437657875232494230, 3.75700289790538821405026449260, 4.71515772321170202464110419962, 5.79594203472025299800211051449, 6.25027473310126323484408443265, 7.13949292316811578804044607765, 7.54842092800110084942822443733
