L(s) = 1 | + (5.05 − 1.18i)3-s + (8.06 + 7.74i)5-s + (−3.75 − 3.75i)7-s + (24.1 − 11.9i)9-s + 6.48i·11-s + (27.4 − 27.4i)13-s + (49.9 + 29.6i)15-s + (−55.8 + 55.8i)17-s − 10.4i·19-s + (−23.4 − 14.5i)21-s + (−115. − 115. i)23-s + (4.99 + 124. i)25-s + (108. − 89.3i)27-s − 267.·29-s − 305.·31-s + ⋯ |
L(s) = 1 | + (0.973 − 0.228i)3-s + (0.721 + 0.692i)5-s + (−0.202 − 0.202i)7-s + (0.895 − 0.444i)9-s + 0.177i·11-s + (0.586 − 0.586i)13-s + (0.860 + 0.509i)15-s + (−0.796 + 0.796i)17-s − 0.126i·19-s + (−0.243 − 0.151i)21-s + (−1.04 − 1.04i)23-s + (0.0399 + 0.999i)25-s + (0.770 − 0.637i)27-s − 1.71·29-s − 1.77·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0184i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.00420 + 0.0184978i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00420 + 0.0184978i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-5.05 + 1.18i)T \) |
| 5 | \( 1 + (-8.06 - 7.74i)T \) |
good | 7 | \( 1 + (3.75 + 3.75i)T + 343iT^{2} \) |
| 11 | \( 1 - 6.48iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-27.4 + 27.4i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (55.8 - 55.8i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 10.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (115. + 115. i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 305.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-187. - 187. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 374. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-291. + 291. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (-230. + 230. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (-178. - 178. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 225.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 315.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-203. - 203. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 161. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-300. + 300. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 948. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (553. + 553. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 577.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-84.9 - 84.9i)T + 9.12e5iT^{2} \) |
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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
−14.59518639568838392187594872682, −13.47635535101467741628331104739, −12.79735075361409108503376522133, −10.91934017875734295554074923141, −9.893981144305214274041221604383, −8.706814928405662176876990717410, −7.34043921618211627083905987932, −6.08710501101516703577556828153, −3.76157364471369755805204728487, −2.13177089886048101420317013605,
2.02323815934912556461247237368, 4.00523781466870002029369537366, 5.71786221426095405602814344957, 7.51145470279425181636527653625, 9.045187447163474348254975627017, 9.441016471256401143629238072236, 11.05598308571421875691882392779, 12.69624490272390843249498149535, 13.56248467793877396471855597176, 14.37830454771524346765994574472