| L(s) = 1 | − 5.65·2-s + (16.9 + 21i)3-s + 32.0·4-s + (−96 − 118. i)6-s − 2i·7-s − 181.·8-s + (−153. + 712. i)9-s + 33.9i·11-s + (543. + 672. i)12-s − 2.95e3i·13-s + 11.3i·14-s + 1.02e3·16-s − 4.48e3·17-s + (865. − 4.03e3i)18-s − 5.25e3·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (0.628 + 0.777i)3-s + 0.500·4-s + (−0.444 − 0.549i)6-s − 0.00583i·7-s − 0.353·8-s + (−0.209 + 0.977i)9-s + 0.0255i·11-s + (0.314 + 0.388i)12-s − 1.34i·13-s + 0.00412i·14-s + 0.250·16-s − 0.911·17-s + (0.148 − 0.691i)18-s − 0.766·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3835191440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3835191440\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5.65T \) |
| 3 | \( 1 + (-16.9 - 21i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 33.9iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.95e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 4.48e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 5.25e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.02e4T + 1.48e8T^{2} \) |
| 29 | \( 1 - 2.20e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.28e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.40e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.67e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.40e3iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.79e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.92e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.26e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 6.25e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.38e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 6.82e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 7.30e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 3.40e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 4.96e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 3.86e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 2.81e5iT - 8.32e11T^{2} \) |
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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
−11.13592276884264140399983945086, −10.40908261852327136685698441205, −9.528054981521024471669711092645, −8.476820434472981727535919426101, −7.77431207257399026891413414500, −6.22888631907273585976186891932, −4.77550169945344862935320375631, −3.34604728106528490016635703666, −2.09561511829983640972034392758, −0.12488629094379360870784103059,
1.50088773631870837053541255559, 2.53506976120086073910876760097, 4.18696767546533948902362758781, 6.26022745896911096392962710220, 6.98875253412587546327836403798, 8.203879704055503590636255930441, 8.933715732852915732468732641250, 9.930439187439537213312791364212, 11.30492156705569290897418825374, 12.09842005966518458860118332205
