| L(s) = 1 | − 2.55·2-s + 1.62·3-s + 4.52·4-s + 1.02·5-s − 4.16·6-s − 2.90·7-s − 6.44·8-s − 0.343·9-s − 2.61·10-s + 3.97·11-s + 7.37·12-s + 13-s + 7.42·14-s + 1.66·15-s + 7.41·16-s + 1.98·17-s + 0.878·18-s − 4.42·19-s + 4.63·20-s − 4.74·21-s − 10.1·22-s + 0.308·23-s − 10.5·24-s − 3.95·25-s − 2.55·26-s − 5.44·27-s − 13.1·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 0.940·3-s + 2.26·4-s + 0.457·5-s − 1.69·6-s − 1.09·7-s − 2.27·8-s − 0.114·9-s − 0.827·10-s + 1.19·11-s + 2.12·12-s + 0.277·13-s + 1.98·14-s + 0.430·15-s + 1.85·16-s + 0.481·17-s + 0.206·18-s − 1.01·19-s + 1.03·20-s − 1.03·21-s − 2.16·22-s + 0.0643·23-s − 2.14·24-s − 0.790·25-s − 0.500·26-s − 1.04·27-s − 2.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 - 1.62T + 3T^{2} \) |
| 5 | \( 1 - 1.02T + 5T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 11 | \( 1 - 3.97T + 11T^{2} \) |
| 17 | \( 1 - 1.98T + 17T^{2} \) |
| 19 | \( 1 + 4.42T + 19T^{2} \) |
| 23 | \( 1 - 0.308T + 23T^{2} \) |
| 29 | \( 1 - 8.91T + 29T^{2} \) |
| 31 | \( 1 - 6.21T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 + 0.0267T + 41T^{2} \) |
| 43 | \( 1 + 9.51T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 6.82T + 59T^{2} \) |
| 61 | \( 1 + 5.60T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 + 2.65T + 71T^{2} \) |
| 73 | \( 1 - 2.32T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−7.935856515109127547121695727085, −6.81887969702261233045085898684, −6.41457998516436649987929540525, −6.01071120174298080580689140509, −4.49345305604229477461733655679, −3.36873300377907739662403644880, −2.91116236807994206607782667996, −2.00714087630705652116298659476, −1.23521599381237245657652636104, 0,
1.23521599381237245657652636104, 2.00714087630705652116298659476, 2.91116236807994206607782667996, 3.36873300377907739662403644880, 4.49345305604229477461733655679, 6.01071120174298080580689140509, 6.41457998516436649987929540525, 6.81887969702261233045085898684, 7.935856515109127547121695727085
