| L(s) = 1 | + (−7.47e4 − 4.31e4i)2-s + (3.41e7 + 2.61e7i)3-s + (1.57e9 + 2.73e9i)4-s + (−9.12e10 − 5.27e10i)5-s + (−1.42e12 − 3.43e12i)6-s + (3.71e12 + 3.30e13i)7-s + (−0.0156 + 9.83e13i)8-s + (4.85e14 + 1.78e15i)9-s + (4.55e15 + 7.88e15i)10-s + (2.11e16 − 1.22e16i)11-s + (−1.75e16 + 1.34e17i)12-s − 1.14e18·13-s + (1.14e18 − 2.62e18i)14-s + (−1.74e18 − 4.18e18i)15-s + (1.10e19 − 1.90e19i)16-s + (−1.82e19 + 1.05e19i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 − 0.658i)2-s + (0.794 + 0.607i)3-s + (0.367 + 0.636i)4-s + (−0.598 − 0.345i)5-s + (−0.505 − 1.21i)6-s + (0.111 + 0.993i)7-s + 0.349i·8-s + (0.261 + 0.965i)9-s + (0.455 + 0.788i)10-s + (0.460 − 0.265i)11-s + (−0.0947 + 0.728i)12-s − 1.72·13-s + (0.526 − 1.20i)14-s + (−0.265 − 0.637i)15-s + (0.597 − 1.03i)16-s + (−0.374 + 0.216i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(0.8272907880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8272907880\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.41e7 - 2.61e7i)T \) |
| 7 | \( 1 + (-3.71e12 - 3.30e13i)T \) |
| good | 2 | \( 1 + (7.47e4 + 4.31e4i)T + (2.14e9 + 3.71e9i)T^{2} \) |
| 5 | \( 1 + (9.12e10 + 5.27e10i)T + (1.16e22 + 2.01e22i)T^{2} \) |
| 11 | \( 1 + (-2.11e16 + 1.22e16i)T + (1.05e33 - 1.82e33i)T^{2} \) |
| 13 | \( 1 + 1.14e18T + 4.42e35T^{2} \) |
| 17 | \( 1 + (1.82e19 - 1.05e19i)T + (1.18e39 - 2.05e39i)T^{2} \) |
| 19 | \( 1 + (3.73e18 - 6.47e18i)T + (-4.15e40 - 7.20e40i)T^{2} \) |
| 23 | \( 1 + (1.71e21 + 9.88e20i)T + (1.88e43 + 3.25e43i)T^{2} \) |
| 29 | \( 1 + 8.94e22iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (1.54e23 + 2.67e23i)T + (-2.64e47 + 4.58e47i)T^{2} \) |
| 37 | \( 1 + (-7.35e24 + 1.27e25i)T + (-7.61e49 - 1.31e50i)T^{2} \) |
| 41 | \( 1 + 9.30e25iT - 4.06e51T^{2} \) |
| 43 | \( 1 + 8.55e25T + 1.86e52T^{2} \) |
| 47 | \( 1 + (-3.70e26 - 2.14e26i)T + (1.60e53 + 2.78e53i)T^{2} \) |
| 53 | \( 1 + (-3.28e27 + 1.89e27i)T + (7.51e54 - 1.30e55i)T^{2} \) |
| 59 | \( 1 + (3.36e27 - 1.94e27i)T + (2.32e56 - 4.02e56i)T^{2} \) |
| 61 | \( 1 + (8.96e26 - 1.55e27i)T + (-6.75e56 - 1.16e57i)T^{2} \) |
| 67 | \( 1 + (-7.26e28 - 1.25e29i)T + (-1.35e58 + 2.35e58i)T^{2} \) |
| 71 | \( 1 - 5.31e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (4.73e29 + 8.19e29i)T + (-2.11e59 + 3.66e59i)T^{2} \) |
| 79 | \( 1 + (1.10e30 - 1.92e30i)T + (-2.64e60 - 4.58e60i)T^{2} \) |
| 83 | \( 1 - 4.18e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (-7.99e30 - 4.61e30i)T + (1.20e62 + 2.07e62i)T^{2} \) |
| 97 | \( 1 - 2.29e31T + 3.77e63T^{2} \) |
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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
−11.47585134750501675242521176277, −10.15434155904364162030124437949, −9.253114043433900167556704561231, −8.514072632520433795925896469580, −7.55941008478351254598059751965, −5.35307335129182762060758838603, −4.13166034625267210816501670803, −2.61046033342356641502743056941, −2.01825756509746881284197069796, −0.39919580013417965731624357000,
0.53975359469832560360592002281, 1.66198881901191968001475343498, 3.17467671379313227560201698982, 4.37297601271034525993082577635, 6.68954729399178922444630494738, 7.33494750209647339064717004111, 7.970688666357512998545484271422, 9.276969718747798022453304023734, 10.15417821134917953128100072074, 11.83115643144109923460401304585
