| L(s) = 1 | − 1.13i·2-s + 3.67i·3-s + 2.70·4-s + 4.18·6-s − 7.62i·8-s − 4.52·9-s + 13.4·11-s + 9.96i·12-s − 20.1i·13-s + 2.16·16-s + 5.14i·18-s + 19·19-s − 15.2i·22-s + 28.0·24-s − 22.8·26-s + 16.4i·27-s + ⋯ |
| L(s) = 1 | − 0.568i·2-s + 1.22i·3-s + 0.677·4-s + 0.696·6-s − 0.953i·8-s − 0.503·9-s + 1.21·11-s + 0.830i·12-s − 1.54i·13-s + 0.135·16-s + 0.285i·18-s + 19-s − 0.693i·22-s + 1.16·24-s − 0.879·26-s + 0.609i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.360902680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.360902680\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 1.13iT - 4T^{2} \) |
| 3 | \( 1 - 3.67iT - 9T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 13.4T + 121T^{2} \) |
| 13 | \( 1 + 20.1iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 73.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 - 69.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 120.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 122. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 185. iT - 9.40e3T^{2} \) |
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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
−10.74415163043389692658855057471, −9.963712260030481825462394408624, −9.485392867295449508750404924309, −8.230830154477521759340914857055, −7.08924158525540651947808698257, −6.00456611408391113748556980956, −4.88285738091654385533365968108, −3.67975665533584596017004600767, −3.00814392324379954726605934782, −1.21444499009614633799792769732,
1.33609675165359832774044985007, 2.26744687574882943749268191217, 3.94092650510769749155984942004, 5.50179049392719711485407496789, 6.62963050789193869507787815318, 6.87682613404549070780503866680, 7.74940677266008655854828029235, 8.776706028748232759019791562003, 9.722197069941727179444206111495, 11.23111497869091078539338833108
