| L(s) = 1 | + (−0.764 + 3.34i)2-s + (−5.99 + 7.51i)3-s + (−3.41 − 1.64i)4-s + (0.968 − 1.21i)5-s + (−20.5 − 25.8i)6-s + (18.5 + 0.839i)7-s + (−9.01 + 11.3i)8-s + (−14.5 − 63.8i)9-s + (3.32 + 4.17i)10-s + (3.27 − 14.3i)11-s + (32.8 − 15.8i)12-s + (−15.5 + 68.1i)13-s + (−16.9 + 61.2i)14-s + (3.32 + 14.5i)15-s + (−49.8 − 62.5i)16-s + (−63.7 + 30.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.270 + 1.18i)2-s + (−1.15 + 1.44i)3-s + (−0.426 − 0.205i)4-s + (0.0866 − 0.108i)5-s + (−1.40 − 1.75i)6-s + (0.998 + 0.0453i)7-s + (−0.398 + 0.499i)8-s + (−0.539 − 2.36i)9-s + (0.105 + 0.131i)10-s + (0.0896 − 0.392i)11-s + (0.790 − 0.380i)12-s + (−0.331 + 1.45i)13-s + (−0.323 + 1.17i)14-s + (0.0572 + 0.250i)15-s + (−0.778 − 0.976i)16-s + (−0.909 + 0.437i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.576i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.234675 - 0.739575i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.234675 - 0.739575i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-18.5 - 0.839i)T \) |
| good | 2 | \( 1 + (0.764 - 3.34i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (5.99 - 7.51i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-0.968 + 1.21i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (-3.27 + 14.3i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (15.5 - 68.1i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (63.7 - 30.6i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 - 87.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + (67.8 + 32.6i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (-85.8 + 41.3i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 - 24.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-118. + 57.0i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (269. - 338. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (5.18 + 6.49i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (-0.791 + 3.46i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (74.5 + 35.9i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (-479. - 601. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-240. + 115. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 730.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-978. - 471. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (2.23 + 9.78i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + 782.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (136. + 598. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (95.1 + 416. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 955.T + 9.12e5T^{2} \) |
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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
−15.98370915375257280114116919693, −15.07674527205013445443496728357, −14.17745629732290176170842937622, −11.71784435722861163244209235445, −11.26074420129230703768522921239, −9.666958046652090176847319474013, −8.572229156088622029099376905271, −6.74184069643093618838520207871, −5.51001615027854399327145920046, −4.44731260483846501247471080861,
0.75789246153369928942320112311, 2.24191985318069339688629656481, 5.25215548173485316380051360735, 6.80011833093220086177639140933, 8.078379514210348471395817293313, 10.20543625656022715115202519868, 11.19088787093793901811596844369, 11.95409573494032162126910809095, 12.74362977384174981342921936412, 13.90123133114954972062210644013
