| L(s) = 1 | + (−1.78e4 + 3.09e4i)2-s + (−2.39e6 − 4.14e6i)3-s + (−3.70e8 − 6.42e8i)4-s + (2.47e9 − 4.28e9i)5-s + 1.71e11·6-s + (−1.35e12 − 1.18e12i)7-s + (7.32e12 + 0.000976i)8-s + (−1.14e13 + 1.98e13i)9-s + (8.84e13 + 1.53e14i)10-s + (−6.36e14 − 1.10e15i)11-s + (−1.77e15 + 3.07e15i)12-s − 1.63e16·13-s + (6.07e16 − 2.07e16i)14-s − 2.36e16·15-s + (6.81e16 − 1.18e17i)16-s + (5.30e17 + 9.19e17i)17-s + ⋯ |
| L(s) = 1 | + (−0.771 + 1.33i)2-s + (−0.288 − 0.499i)3-s + (−0.690 − 1.19i)4-s + (0.181 − 0.313i)5-s + 0.890·6-s + (−0.753 − 0.657i)7-s + 0.588·8-s + (−0.166 + 0.288i)9-s + (0.279 + 0.484i)10-s + (−0.505 − 0.874i)11-s + (−0.398 + 0.690i)12-s − 1.14·13-s + (1.46 − 0.499i)14-s − 0.209·15-s + (0.236 − 0.409i)16-s + (0.764 + 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.3928529228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3928529228\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.39e6 + 4.14e6i)T \) |
| 7 | \( 1 + (1.35e12 + 1.18e12i)T \) |
| good | 2 | \( 1 + (1.78e4 - 3.09e4i)T + (-2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-2.47e9 + 4.28e9i)T + (-9.31e19 - 1.61e20i)T^{2} \) |
| 11 | \( 1 + (6.36e14 + 1.10e15i)T + (-7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 + 1.63e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (-5.30e17 - 9.19e17i)T + (-2.40e35 + 4.17e35i)T^{2} \) |
| 19 | \( 1 + (4.72e17 - 8.17e17i)T + (-6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (3.37e19 - 5.85e19i)T + (-1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 + 2.75e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-3.27e21 - 5.66e21i)T + (-8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (3.84e22 - 6.66e22i)T + (-1.50e45 - 2.60e45i)T^{2} \) |
| 41 | \( 1 - 2.49e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 5.09e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-7.78e23 + 1.34e24i)T + (-1.54e48 - 2.68e48i)T^{2} \) |
| 53 | \( 1 + (9.52e24 + 1.64e25i)T + (-5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (4.66e24 + 8.07e24i)T + (-1.13e51 + 1.95e51i)T^{2} \) |
| 61 | \( 1 + (3.38e25 - 5.86e25i)T + (-2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (2.42e26 + 4.20e26i)T + (-4.52e52 + 7.82e52i)T^{2} \) |
| 71 | \( 1 - 1.48e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (2.72e26 + 4.71e26i)T + (-5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-6.93e26 + 1.20e27i)T + (-5.37e54 - 9.30e54i)T^{2} \) |
| 83 | \( 1 + 1.70e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (2.59e27 - 4.48e27i)T + (-1.70e56 - 2.95e56i)T^{2} \) |
| 97 | \( 1 - 7.84e28T + 4.13e57T^{2} \) |
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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
−12.33274423583685181971544169982, −10.49417256249585818238807221736, −9.408123503790035820480212337959, −8.081769483740804481902571374664, −7.28194731176638871052629927572, −6.14530305697161528220323061970, −5.30092273441692937135038944349, −3.33593732832490095914456835193, −1.40230300730717672067262660526, −0.22681034679283609798200705354,
0.50518837189533502847921984737, 2.35678164778390654192529966651, 2.73462232640869356158877402694, 4.32724313840797460226582556598, 5.81190047270661259517099741862, 7.49892550059839496471919876062, 9.281797014636054014898490718313, 9.761531544720039255586443629815, 10.74590671904224777702177010186, 12.04920341876141856604831833280
