| L(s) = 1 | + (−1.39e4 − 2.41e4i)2-s + (2.39e6 − 4.14e6i)3-s + (−1.20e8 + 2.08e8i)4-s + (−1.12e10 − 1.94e10i)5-s − 1.33e11·6-s + (1.74e12 + 4.05e11i)7-s + (−8.25e12 − 0.000976i)8-s + (−1.14e13 − 1.98e13i)9-s + (−3.13e14 + 5.43e14i)10-s + (3.46e14 − 6.00e14i)11-s + (5.76e14 + 9.98e14i)12-s − 1.33e16·13-s + (−1.45e16 − 4.78e16i)14-s − 1.07e17·15-s + (1.79e17 + 3.11e17i)16-s + (6.20e16 − 1.07e17i)17-s + ⋯ |
| L(s) = 1 | + (−0.601 − 1.04i)2-s + (0.288 − 0.499i)3-s + (−0.224 + 0.388i)4-s + (−0.823 − 1.42i)5-s − 0.694·6-s + (0.974 + 0.225i)7-s − 0.663·8-s + (−0.166 − 0.288i)9-s + (−0.991 + 1.71i)10-s + (0.275 − 0.476i)11-s + (0.129 + 0.224i)12-s − 0.943·13-s + (−0.350 − 1.15i)14-s − 0.951·15-s + (0.623 + 1.08i)16-s + (0.0894 − 0.154i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.7089062880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7089062880\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (-1.74e12 - 4.05e11i)T \) |
| good | 2 | \( 1 + (1.39e4 + 2.41e4i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (1.12e10 + 1.94e10i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-3.46e14 + 6.00e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 + 1.33e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (-6.20e16 + 1.07e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (-5.53e17 - 9.58e17i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-1.31e19 - 2.26e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 - 1.08e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (1.80e21 - 3.11e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (2.19e22 + 3.79e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 - 3.88e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 4.95e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-1.12e24 - 1.95e24i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (9.50e24 - 1.64e25i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (3.99e25 - 6.91e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-8.22e24 - 1.42e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (1.51e26 - 2.62e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 - 1.19e27T + 4.85e53T^{2} \) |
| 73 | \( 1 + (1.86e26 - 3.22e26i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-1.28e27 - 2.21e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 + 6.06e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (-2.38e27 - 4.13e27i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 + 1.14e29T + 4.13e57T^{2} \) |
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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
−12.03980379298809035269070566901, −11.02242406083461636320234207703, −9.330573596809599728239908079902, −8.572086415396537179465783847958, −7.60222805685716976076916363157, −5.52410864655941158634987165195, −4.25976560723257079429095438193, −2.78157328107353069362323136629, −1.46979031300239239937724560336, −0.925663695585463360546775283257,
0.21439253264707799547818228317, 2.34551238574930339091773667111, 3.52203859289151628541701767776, 4.87263064215356409741777530918, 6.61281080769708099105192466271, 7.46679217027322463902486209574, 8.216830761996035154416930188595, 9.698713167244102620000494473211, 10.92972140927122156930902986427, 11.98270510789966948745460335932
