| L(s) = 1 | + (4.15 − 3.11i)3-s + (−4.98 + 8.63i)5-s + (−18.3 + 2.18i)7-s + (7.57 − 25.9i)9-s + (12.5 + 21.7i)11-s + (32.7 + 56.6i)13-s + (6.18 + 51.4i)15-s + (2.49 − 4.31i)17-s + (57.9 + 100. i)19-s + (−69.6 + 66.3i)21-s + (−30.3 + 52.5i)23-s + (12.8 + 22.2i)25-s + (−49.2 − 131. i)27-s + (−109. + 190. i)29-s + 139.·31-s + ⋯ |
| L(s) = 1 | + (0.800 − 0.599i)3-s + (−0.445 + 0.772i)5-s + (−0.993 + 0.117i)7-s + (0.280 − 0.959i)9-s + (0.344 + 0.596i)11-s + (0.697 + 1.20i)13-s + (0.106 + 0.885i)15-s + (0.0355 − 0.0615i)17-s + (0.699 + 1.21i)19-s + (−0.723 + 0.689i)21-s + (−0.275 + 0.476i)23-s + (0.102 + 0.177i)25-s + (−0.351 − 0.936i)27-s + (−0.703 + 1.21i)29-s + 0.806·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.744364764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.744364764\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.15 + 3.11i)T \) |
| 7 | \( 1 + (18.3 - 2.18i)T \) |
| good | 5 | \( 1 + (4.98 - 8.63i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-12.5 - 21.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-32.7 - 56.6i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-2.49 + 4.31i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-57.9 - 100. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (30.3 - 52.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (109. - 190. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (162. + 282. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-69.8 - 120. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-68.4 + 118. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 140.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (356. - 617. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + 475.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 565.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 200.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-224. + 388. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + 374.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (1.50 - 2.60i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-189. - 328. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-757. + 1.31e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.02486411538223809056173314202, −10.84452144879379702446474293265, −9.600684328838555610979948098727, −9.004246185063170160686972022124, −7.62828194654138937498638538867, −6.95289494214951986004266504669, −6.06153723316813346779754802012, −3.93670385860902850924001586048, −3.17131435388965836063359126702, −1.64792560206125858866075461008,
0.64957286709115475016863450967, 2.88958913894615433755124629833, 3.80800660229250376371806782840, 5.00623878311450129936366433540, 6.33581473322941272928512825987, 7.80898171972478412818922288056, 8.565063233885029910034905383075, 9.406953511884168363251448875912, 10.29250196303142690255101067182, 11.33430346351992548641712757617
