| L(s) = 1 | − 9.27i·2-s − 5.85i·3-s − 54.1·4-s − 54.3·6-s + 90.4i·7-s + 205. i·8-s + 208.·9-s + 597.·11-s + 316. i·12-s + 169i·13-s + 838.·14-s + 172.·16-s − 1.34e3i·17-s − 1.93e3i·18-s − 2.98e3·19-s + ⋯ |
| L(s) = 1 | − 1.64i·2-s − 0.375i·3-s − 1.69·4-s − 0.616·6-s + 0.697i·7-s + 1.13i·8-s + 0.858·9-s + 1.48·11-s + 0.635i·12-s + 0.277i·13-s + 1.14·14-s + 0.168·16-s − 1.13i·17-s − 1.40i·18-s − 1.89·19-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
| \(L(3)\) |
\(\approx\) |
\(1.256524964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256524964\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 9.27iT - 32T^{2} \) |
| 3 | \( 1 + 5.85iT - 243T^{2} \) |
| 7 | \( 1 - 90.4iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 597.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.34e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.98e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.35e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 6.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 661.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.68e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 6.93e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.51e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.02e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.06e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 7.02e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.12e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.78e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.90e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.90e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 5.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.54e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 7.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.92e4iT - 8.58e9T^{2} \) |
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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
−10.31896644427148762774946748097, −9.238248359584698173594972129510, −8.887723291467197565099059070505, −7.24367685040386872336981404468, −6.20203936037960732801638984267, −4.55585699319447266713161820317, −3.81229781654973744883888249089, −2.35100940510294265357330748512, −1.62243781023993130175905777159, −0.33546296109470443883672761727,
1.45403863593380759024457359846, 3.98831387032827251861588781562, 4.34563625381918949397330658723, 5.84751013555630590043114248937, 6.62780367512104844096366159538, 7.39310387972771777917244494574, 8.423284365032894678026106216228, 9.253714814705280606041133038009, 10.22239468723216846865215957894, 11.21927522070501832182885933930
