L(s) = 1 | + (−2.28 − 3.96i)2-s + (−3.36 + 3.96i)3-s + (−6.45 + 11.1i)4-s + (23.3 + 4.26i)6-s + (−10.0 − 17.4i)7-s + 22.4·8-s + (−4.37 − 26.6i)9-s + (−33.1 − 57.4i)11-s + (−22.5 − 63.2i)12-s + (−23.4 + 40.5i)13-s + (−45.9 + 79.6i)14-s + (0.237 + 0.411i)16-s + 47.6·17-s + (−95.5 + 78.2i)18-s − 9.95·19-s + ⋯ |
L(s) = 1 | + (−0.808 − 1.40i)2-s + (−0.647 + 0.762i)3-s + (−0.807 + 1.39i)4-s + (1.59 + 0.290i)6-s + (−0.543 − 0.940i)7-s + 0.993·8-s + (−0.162 − 0.986i)9-s + (−0.909 − 1.57i)11-s + (−0.543 − 1.52i)12-s + (−0.499 + 0.864i)13-s + (−0.878 + 1.52i)14-s + (0.00371 + 0.00643i)16-s + 0.679·17-s + (−1.25 + 1.02i)18-s − 0.120·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.294840 + 0.0502080i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.294840 + 0.0502080i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (3.36 - 3.96i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (2.28 + 3.96i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (10.0 + 17.4i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (33.1 + 57.4i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (23.4 - 40.5i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 47.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 9.95T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-4.79 + 8.30i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-89.3 - 154. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (77.0 - 133. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 248.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (124. - 216. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (106. + 183. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-237. - 411. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 546.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-209. + 363. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-272. - 472. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (223. - 387. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 409.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 358.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-325. - 564. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (406. + 704. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 200.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-126. - 218. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.54251073669190235268995163896, −10.57888377435131675848121268268, −10.35658406912135711714443801951, −9.277933609410937043342933845763, −8.393751307887882064283060239565, −6.83495794378899783409763730455, −5.38143632498261329945670915146, −3.87390192234261515494197132685, −3.02319003661074136475459149154, −0.915041730627423745209140239831,
0.23413845665253988881949403473, 2.35840736866336011463408864364, 5.14328922242240772129558434686, 5.71847167918843917497419534861, 6.87703522525078749458995727091, 7.57865521139392017500206551432, 8.399678956758976257842353892285, 9.711262260817522990295617917859, 10.34334291867878602265615036077, 12.07948584025783279758560037653