L(s) = 1 | + 1.58·2-s + 4.98i·3-s − 5.49·4-s + 14.4i·5-s + 7.88i·6-s + 29.4i·7-s − 21.3·8-s + 2.16·9-s + 22.8i·10-s − 13.5i·11-s − 27.4i·12-s + 45.7·13-s + 46.5i·14-s − 72.0·15-s + 10.2·16-s + ⋯ |
L(s) = 1 | + 0.559·2-s + 0.959i·3-s − 0.687·4-s + 1.29i·5-s + 0.536i·6-s + 1.58i·7-s − 0.943·8-s + 0.0800·9-s + 0.722i·10-s − 0.370i·11-s − 0.659i·12-s + 0.976·13-s + 0.888i·14-s − 1.23·15-s + 0.159·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.526518186\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526518186\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 - 1.58T + 8T^{2} \) |
| 3 | \( 1 - 4.98iT - 27T^{2} \) |
| 5 | \( 1 - 14.4iT - 125T^{2} \) |
| 7 | \( 1 - 29.4iT - 343T^{2} \) |
| 11 | \( 1 + 13.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 45.7T + 2.19e3T^{2} \) |
| 19 | \( 1 + 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 144. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 1.26iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 30.0iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 398. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 184. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 135.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 247.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 635.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 625.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 166. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 159.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 19.4iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 336. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 47.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 626.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 692. iT - 9.12e5T^{2} \) |
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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
−11.87136118311497550232562688118, −10.81660454747673837930706965505, −10.13785290018154918275349178628, −8.957187497344319346439975446123, −8.468434223105362237465561275974, −6.53605736689364904847291644653, −5.82461576467666298398880649703, −4.64578859211818381377610801725, −3.58980028228867299473614893517, −2.59322514157911689304347938675,
0.51966383622354098495824306191, 1.48289000723684138876471632601, 3.88139137240546840031022016946, 4.45235228617566331343022959884, 5.73116266586303545439091195424, 6.92748018831700170709072418072, 7.931948694822967481211866689628, 8.800216746034006396931794756380, 9.832730237908134618400379933058, 10.99529400924574309243665532609