| L(s) = 1 | + 3.78i·2-s + 8.20i·3-s + 17.6·4-s − 31.0·6-s − 88.9i·7-s + 188. i·8-s + 175.·9-s − 156.·11-s + 145. i·12-s + 169i·13-s + 336.·14-s − 145.·16-s − 447. i·17-s + 664. i·18-s + 269.·19-s + ⋯ |
| L(s) = 1 | + 0.668i·2-s + 0.526i·3-s + 0.552·4-s − 0.352·6-s − 0.686i·7-s + 1.03i·8-s + 0.722·9-s − 0.390·11-s + 0.290i·12-s + 0.277i·13-s + 0.459·14-s − 0.142·16-s − 0.375i·17-s + 0.483i·18-s + 0.171·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.625025574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.625025574\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 2 | \( 1 - 3.78iT - 32T^{2} \) |
| 3 | \( 1 - 8.20iT - 243T^{2} \) |
| 7 | \( 1 + 88.9iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 156.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 447. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 269.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.37e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 797.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.39e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.43e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.13e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 9.97e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.15e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.13e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.21e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.30e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.54e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 5.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.73e4iT - 8.58e9T^{2} \) |
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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
−10.92478876659734288405772046629, −10.21282391674200342950759019979, −9.244983650484167423809243783536, −7.932439714749123117678739107948, −7.26330665869461945153923952714, −6.35760708581194387992497164088, −5.15543729445058983672468479508, −4.17315358991876740513198613666, −2.77706767818221027446723659999, −1.24971763256407764072469099259,
0.72195375650107180667651601060, 1.92290755612145368656059573563, 2.79319066634405293823635598369, 4.14834568141101641153408609077, 5.64230680119232670955078797939, 6.66448648677402000510918547861, 7.51285482202065913064276508399, 8.583251504434373456831021302036, 9.833681550909324982747309189538, 10.52170641979929578500788896811
