L(s) = 1 | + (−0.582 − 2.17i)2-s + (−1.71 − 0.245i)3-s + (−2.64 + 1.52i)4-s + (0.465 + 3.86i)6-s + (1.15 + 2.38i)7-s + (1.68 + 1.68i)8-s + (2.87 + 0.840i)9-s + (3.88 − 2.24i)11-s + (4.91 − 1.97i)12-s + (1.08 − 1.08i)13-s + (4.50 − 3.88i)14-s + (−0.381 + 0.660i)16-s + (2.04 + 0.548i)17-s + (0.150 − 6.74i)18-s + (−3.66 − 2.11i)19-s + ⋯ |
L(s) = 1 | + (−0.411 − 1.53i)2-s + (−0.989 − 0.141i)3-s + (−1.32 + 0.764i)4-s + (0.189 + 1.57i)6-s + (0.435 + 0.900i)7-s + (0.595 + 0.595i)8-s + (0.959 + 0.280i)9-s + (1.17 − 0.676i)11-s + (1.41 − 0.569i)12-s + (0.300 − 0.300i)13-s + (1.20 − 1.03i)14-s + (−0.0952 + 0.165i)16-s + (0.496 + 0.133i)17-s + (0.0354 − 1.58i)18-s + (−0.839 − 0.484i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.361098 - 0.782170i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.361098 - 0.782170i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.71 + 0.245i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.15 - 2.38i)T \) |
good | 2 | \( 1 + (0.582 + 2.17i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (-3.88 + 2.24i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.08 + 1.08i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.04 - 0.548i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (3.66 + 2.11i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.13 + 0.840i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 + (-0.530 - 0.918i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.75 + 1.54i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 5.84iT - 41T^{2} \) |
| 43 | \( 1 + (2.00 - 2.00i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.36 + 5.10i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.23 + 8.34i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.35 - 4.07i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.88 + 6.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.152 + 0.569i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 4.66iT - 71T^{2} \) |
| 73 | \( 1 + (-4.22 - 1.13i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.78 - 3.33i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (11.0 + 11.0i)T + 83iT^{2} \) |
| 89 | \( 1 + (-1.75 + 3.04i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.60 - 5.60i)T + 97iT^{2} \) |
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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
−10.86009897645038745845979376943, −9.907781062322660397522497948643, −8.995260911698862891264506619996, −8.297597504799546642864798023392, −6.71745179920193953066605738293, −5.81365365520453896238301981401, −4.63220283755806403076042215573, −3.48742778613091361243970165454, −2.07098496844522106136437249673, −0.856829014403081362092767695709,
1.17470355447169958480115751218, 4.09017285716180243190290257374, 4.78682950719566872660021251764, 5.95154864327483528980860186984, 6.68741824039036327821637625752, 7.29846097890232963476349591310, 8.251546142559631494381838697933, 9.371066918027060007564464464268, 10.05776976283286997965563954073, 11.12229783820986921121228294214