L(s) = 1 | + (−1 + 1.73i)2-s + (−1.5 − 2.59i)3-s + (−1.99 − 3.46i)4-s + (−1 + 1.73i)5-s + 6·6-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (−1.99 − 3.46i)10-s + (4 + 6.92i)11-s + (−6.00 + 10.3i)12-s − 42·13-s + 6·15-s + (−8 + 13.8i)16-s + (1 + 1.73i)17-s + (−9 − 15.5i)18-s + (62 − 107. i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.0894 + 0.154i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.0632 − 0.109i)10-s + (0.109 + 0.189i)11-s + (−0.144 + 0.249i)12-s − 0.896·13-s + 0.103·15-s + (−0.125 + 0.216i)16-s + (0.0142 + 0.0247i)17-s + (−0.117 − 0.204i)18-s + (0.748 − 1.29i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.145887577\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145887577\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 + (1.5 + 2.59i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-4 - 6.92i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 42T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-62 + 107. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (38 - 65.8i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 254T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-36 - 62.3i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (199 - 344. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 462T + 6.89e4T^{2} \) |
| 43 | \( 1 - 212T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-132 + 228. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-81 - 140. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-386 - 668. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (15 - 25.9i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-382 - 661. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 236T + 3.57e5T^{2} \) |
| 73 | \( 1 + (209 + 361. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (276 - 478. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (15 - 25.9i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
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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
−11.52727463213481976172045704171, −10.44686246803069579045987689704, −9.500653316873998047545419363587, −8.509382046497376126155616272364, −7.34712297921360505520584870228, −6.87328990228092010242518390699, −5.59358524038180540168885129591, −4.61955876626403286174948777698, −2.74173493525254529146793396765, −0.986466579720215026231979625906,
0.66895836114772020108419299615, 2.47697824517564855148506211167, 3.85206920583382157080152735104, 4.89388587069008611216404283598, 6.14297698309777683964648137273, 7.53714887398862525827833954880, 8.511891307323461194799340052919, 9.551528381142923962546914405606, 10.25539281831255275006967577502, 11.07443766045306164027814323012