L(s) = 1 | + (5.42 + 9.40i)5-s + (18.2 − 3.24i)7-s + (−44.9 − 25.9i)11-s + 32.1i·13-s + (40.7 − 70.5i)17-s + (0.0420 − 0.0242i)19-s + (77.3 − 44.6i)23-s + (3.58 − 6.20i)25-s − 175. i·29-s + (−186. − 107. i)31-s + (129. + 153. i)35-s + (−32.2 − 55.8i)37-s + 411.·41-s + 234.·43-s + (−316. − 547. i)47-s + ⋯ |
L(s) = 1 | + (0.485 + 0.840i)5-s + (0.984 − 0.175i)7-s + (−1.23 − 0.710i)11-s + 0.686i·13-s + (0.581 − 1.00i)17-s + (0.000507 − 0.000293i)19-s + (0.701 − 0.404i)23-s + (0.0286 − 0.0496i)25-s − 1.12i·29-s + (−1.07 − 0.622i)31-s + (0.625 + 0.742i)35-s + (−0.143 − 0.248i)37-s + 1.56·41-s + 0.832·43-s + (−0.980 − 1.69i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.281931787\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.281931787\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-18.2 + 3.24i)T \) |
good | 5 | \( 1 + (-5.42 - 9.40i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (44.9 + 25.9i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 32.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-40.7 + 70.5i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-0.0420 + 0.0242i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-77.3 + 44.6i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 175. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (186. + 107. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (32.2 + 55.8i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 411.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 234.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (316. + 547. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-230. - 132. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (175. - 304. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (673. - 389. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (98.0 - 169. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 142. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-676. - 390. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-644. - 1.11e3i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 235.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (335. + 580. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 655. iT - 9.12e5T^{2} \) |
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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
−9.571600509001747050089261691926, −8.637665354136374593607658272555, −7.69254919822448900954426065738, −7.13753019169304980167411359260, −5.96992400240660731650806026142, −5.26580192532818638394183856020, −4.22613856262248396264849638953, −2.89249997655425083067365170599, −2.15797155271219760252346593082, −0.62061449709242920647395653309,
1.11731389101779815269325248988, 2.00177056284670617495097603608, 3.29403130521967617259205011650, 4.78283580844198334506796873485, 5.16581873672954278690587954868, 5.99825761404502298812486185686, 7.49222651660887857544060427616, 7.916040280768081325363453108488, 8.869197520310336405307702606708, 9.552517037929738751971410037071