| L(s) = 1 | + (0.674 + 0.490i)2-s + (0.345 + 3.28i)3-s + (−0.403 − 1.24i)4-s + (−1.37 + 2.38i)6-s + (−4.81 − 1.02i)7-s + (0.851 − 2.62i)8-s + (−7.73 + 1.64i)9-s + (0.421 − 0.467i)11-s + (3.93 − 1.75i)12-s + (−1.38 − 0.618i)13-s + (−2.74 − 3.05i)14-s + (−0.250 + 0.181i)16-s + (−0.729 − 0.810i)17-s + (−6.02 − 2.68i)18-s + (3.01 − 1.34i)19-s + ⋯ |
| L(s) = 1 | + (0.477 + 0.346i)2-s + (0.199 + 1.89i)3-s + (−0.201 − 0.620i)4-s + (−0.562 + 0.974i)6-s + (−1.81 − 0.386i)7-s + (0.301 − 0.926i)8-s + (−2.57 + 0.548i)9-s + (0.127 − 0.141i)11-s + (1.13 − 0.505i)12-s + (−0.385 − 0.171i)13-s + (−0.734 − 0.815i)14-s + (−0.0626 + 0.0454i)16-s + (−0.176 − 0.196i)17-s + (−1.42 − 0.632i)18-s + (0.691 − 0.308i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0208428 - 0.0251772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0208428 - 0.0251772i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + (5.25 - 1.83i)T \) |
| good | 2 | \( 1 + (-0.674 - 0.490i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.345 - 3.28i)T + (-2.93 + 0.623i)T^{2} \) |
| 7 | \( 1 + (4.81 + 1.02i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (-0.421 + 0.467i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (1.38 + 0.618i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (0.729 + 0.810i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.01 + 1.34i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.450 - 1.38i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (4.03 + 2.93i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (2.60 - 4.50i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.498 - 4.74i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (0.887 - 0.395i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-4.61 + 3.35i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (8.26 - 1.75i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.272 + 2.59i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + 6.88T + 61T^{2} \) |
| 67 | \( 1 + (-0.697 - 1.20i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.22 - 1.53i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (6.40 - 7.11i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-8.36 - 9.28i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.677 + 6.44i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (3.08 + 9.48i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.66 + 5.12i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.863661392914917706619846170984, −9.564611305384804929033569596456, −8.855239754540240450214391201653, −7.27851068206478775767950426772, −6.19538384142262732155490860948, −5.49578539043034544902989425547, −4.58266830048830358760043107660, −3.69785148302204469853688958305, −3.03315392019670828381022351716, −0.01271852580819020525614296749,
1.98186366287975944078296168597, 2.91401796597454664905019743970, 3.65808352075744860433588695873, 5.47894383114356508115709071541, 6.28912422655028157488722823645, 7.16544731019943973126243804909, 7.68743400300805756122048312862, 8.863650421916143738281351661242, 9.335790876686817249083304270975, 10.84589973287074342875707820469
