L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (−8.97 + 15.5i)5-s + (13.2 − 12.9i)7-s − 7.99·8-s + (17.9 + 31.0i)10-s + (−19.8 − 34.3i)11-s + 28.4·13-s + (−9.22 − 35.8i)14-s + (−8 + 13.8i)16-s + (7.38 + 12.7i)17-s + (−33.9 + 58.7i)19-s + 71.8·20-s − 79.3·22-s + (−88.8 + 153. i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.802 + 1.39i)5-s + (0.714 − 0.699i)7-s − 0.353·8-s + (0.567 + 0.983i)10-s + (−0.543 − 0.941i)11-s + 0.606·13-s + (−0.176 − 0.684i)14-s + (−0.125 + 0.216i)16-s + (0.105 + 0.182i)17-s + (−0.409 + 0.709i)19-s + 0.802·20-s − 0.769·22-s + (−0.805 + 1.39i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2060909064\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2060909064\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-13.2 + 12.9i)T \) |
good | 5 | \( 1 + (8.97 - 15.5i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (19.8 + 34.3i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 28.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-7.38 - 12.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (33.9 - 58.7i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (88.8 - 153. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 209.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (143. + 249. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (183. - 318. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 294.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 348.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (119. - 206. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (183. + 318. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-127. - 221. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-149. + 258. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-499. - 864. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 659.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (319. + 553. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-338. + 586. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 830.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (231. - 401. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 373.T + 9.12e5T^{2} \) |
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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
−11.33512341346874615712321666776, −10.65732544635163499072494182677, −9.889230674343010214946608532345, −8.260182476995878583878485417209, −7.67678419136027767063057006565, −6.50081312319971138268126958898, −5.40032708900077772523009524258, −3.80220911749361987296092249267, −3.43835058544111494226552412324, −1.80022967933081745604693068960,
0.06000680351829049141800849793, 1.92274628572807976164754840528, 3.83115524516218526794955051737, 4.88009240540968780191196017099, 5.33298926340065127559043439570, 6.84419761495541925600992016544, 7.948437401872367788115699384135, 8.541431291048121058225985720357, 9.239902153782398328992412346727, 10.73470220963338770741407737037