| L(s) = 1 | + (−1.17 − 4.36i)2-s + (2.89 + 0.776i)3-s + (−10.7 + 6.22i)4-s + (−6.73 + 8.92i)5-s − 13.5i·6-s + (−1.57 + 18.4i)7-s + (14.1 + 14.1i)8-s + (7.79 + 4.50i)9-s + (46.8 + 18.9i)10-s + (19.3 + 33.4i)11-s + (−36.0 + 9.66i)12-s + (−34.4 + 34.4i)13-s + (82.4 − 14.7i)14-s + (−26.4 + 20.6i)15-s + (−4.37 + 7.57i)16-s + (20.7 − 77.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.413 − 1.54i)2-s + (0.557 + 0.149i)3-s + (−1.34 + 0.777i)4-s + (−0.602 + 0.798i)5-s − 0.922i·6-s + (−0.0849 + 0.996i)7-s + (0.627 + 0.627i)8-s + (0.288 + 0.166i)9-s + (1.48 + 0.599i)10-s + (0.529 + 0.916i)11-s + (−0.867 + 0.232i)12-s + (−0.735 + 0.735i)13-s + (1.57 − 0.281i)14-s + (−0.455 + 0.355i)15-s + (−0.0683 + 0.118i)16-s + (0.296 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.935770 + 0.199942i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.935770 + 0.199942i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.89 - 0.776i)T \) |
| 5 | \( 1 + (6.73 - 8.92i)T \) |
| 7 | \( 1 + (1.57 - 18.4i)T \) |
| good | 2 | \( 1 + (1.17 + 4.36i)T + (-6.92 + 4i)T^{2} \) |
| 11 | \( 1 + (-19.3 - 33.4i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (34.4 - 34.4i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (-20.7 + 77.5i)T + (-4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (41.7 - 72.2i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-50.8 + 13.6i)T + (1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + 83.4iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (81.2 - 46.8i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-42.2 - 157. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 - 412. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-31.4 - 31.4i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (406. - 108. i)T + (8.99e4 - 5.19e4i)T^{2} \) |
| 53 | \( 1 + (-146. + 548. i)T + (-1.28e5 - 7.44e4i)T^{2} \) |
| 59 | \( 1 + (-280. - 485. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-204. - 118. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-956. - 256. i)T + (2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + 738.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-608. - 163. i)T + (3.36e5 + 1.94e5i)T^{2} \) |
| 79 | \( 1 + (153. + 88.4i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-712. + 712. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + (-489. + 847. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.26e3 + 1.26e3i)T + 9.12e5iT^{2} \) |
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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
−12.92422250627704498700943830931, −11.93272604786198539065375350376, −11.46991142829180370312092674595, −10.00531876040197279605345986995, −9.477953920596821844880073898061, −8.261346957181648024674282714215, −6.82432202755272489738043302355, −4.47339079278431425903001671009, −3.11653464447195108469270446929, −2.07149426921607708032762394268,
0.57719571682383436869485206286, 3.82217831836617399824079556692, 5.27332263579917546079682306451, 6.75658964653972720071400084182, 7.72660539418276922943762210187, 8.483864417109422016643715430109, 9.402577896842606212158541198879, 10.89662419372034837896308453979, 12.55655111792780084616662061022, 13.51376567451027379671916790127
