| L(s) = 1 | − 2.50·2-s − 3-s + 4.25·4-s + 5-s + 2.50·6-s + 4.30·7-s − 5.63·8-s + 9-s − 2.50·10-s + 2.03·11-s − 4.25·12-s − 3.82·13-s − 10.7·14-s − 15-s + 5.58·16-s − 5.83·17-s − 2.50·18-s − 3.96·19-s + 4.25·20-s − 4.30·21-s − 5.08·22-s + 7.94·23-s + 5.63·24-s + 25-s + 9.56·26-s − 27-s + 18.2·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 0.577·3-s + 2.12·4-s + 0.447·5-s + 1.02·6-s + 1.62·7-s − 1.99·8-s + 0.333·9-s − 0.790·10-s + 0.612·11-s − 1.22·12-s − 1.06·13-s − 2.87·14-s − 0.258·15-s + 1.39·16-s − 1.41·17-s − 0.589·18-s − 0.908·19-s + 0.951·20-s − 0.938·21-s − 1.08·22-s + 1.65·23-s + 1.15·24-s + 0.200·25-s + 1.87·26-s − 0.192·27-s + 3.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 + 5.83T + 17T^{2} \) |
| 19 | \( 1 + 3.96T + 19T^{2} \) |
| 23 | \( 1 - 7.94T + 23T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 - 7.99T + 37T^{2} \) |
| 41 | \( 1 + 7.66T + 41T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 + 8.02T + 47T^{2} \) |
| 53 | \( 1 - 8.28T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 6.76T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 4.33T + 71T^{2} \) |
| 73 | \( 1 + 0.442T + 73T^{2} \) |
| 79 | \( 1 + 4.50T + 79T^{2} \) |
| 83 | \( 1 - 0.380T + 83T^{2} \) |
| 89 | \( 1 + 7.02T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.77457891922662065865627689100, −7.16751731870975048382757627389, −6.68456490789614388896514832356, −5.77026974745208868219124290879, −4.89467532515602456075377479534, −4.27626861485508329539626375126, −2.58669011186897698470356484721, −1.90253229239602930329937091228, −1.24329632914300978355211734651, 0,
1.24329632914300978355211734651, 1.90253229239602930329937091228, 2.58669011186897698470356484721, 4.27626861485508329539626375126, 4.89467532515602456075377479534, 5.77026974745208868219124290879, 6.68456490789614388896514832356, 7.16751731870975048382757627389, 7.77457891922662065865627689100
