| L(s) = 1 | + (2.03e4 − 3.52e4i)2-s + (2.39e6 + 4.14e6i)3-s + (−5.58e8 − 9.66e8i)4-s + (9.23e9 − 1.59e10i)5-s + 1.94e11·6-s + (−1.57e12 + 8.56e11i)7-s + (−2.35e13 − 0.00781i)8-s + (−1.14e13 + 1.98e13i)9-s + (−3.75e14 − 6.50e14i)10-s + (9.06e14 + 1.57e15i)11-s + (2.66e15 − 4.62e15i)12-s + 2.11e16·13-s + (−1.87e15 + 7.29e16i)14-s + 8.83e16·15-s + (−1.79e17 + 3.10e17i)16-s + (5.75e17 + 9.97e17i)17-s + ⋯ |
| L(s) = 1 | + (0.877 − 1.51i)2-s + (0.288 + 0.499i)3-s + (−1.03 − 1.80i)4-s + (0.676 − 1.17i)5-s + 1.01·6-s + (−0.878 + 0.477i)7-s − 1.89·8-s + (−0.166 + 0.288i)9-s + (−1.18 − 2.05i)10-s + (0.720 + 1.24i)11-s + (0.600 − 1.03i)12-s + 1.49·13-s + (−0.0452 + 1.75i)14-s + 0.781·15-s + (−0.622 + 1.07i)16-s + (0.829 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.884i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(4.880294612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.880294612\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (1.57e12 - 8.56e11i)T \) |
| good | 2 | \( 1 + (-2.03e4 + 3.52e4i)T + (-2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-9.23e9 + 1.59e10i)T + (-9.31e19 - 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-9.06e14 - 1.57e15i)T + (-7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 - 2.11e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (-5.75e17 - 9.97e17i)T + (-2.40e35 + 4.17e35i)T^{2} \) |
| 19 | \( 1 + (-2.82e18 + 4.88e18i)T + (-6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-1.41e19 + 2.45e19i)T + (-1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 - 6.27e20T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-2.07e21 - 3.59e21i)T + (-8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-3.15e22 + 5.46e22i)T + (-1.50e45 - 2.60e45i)T^{2} \) |
| 41 | \( 1 + 2.01e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 8.48e22T + 2.34e47T^{2} \) |
| 47 | \( 1 + (1.30e24 - 2.26e24i)T + (-1.54e48 - 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-3.05e24 - 5.28e24i)T + (-5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-2.97e25 - 5.16e25i)T + (-1.13e51 + 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-1.55e25 + 2.68e25i)T + (-2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (1.52e26 + 2.64e26i)T + (-4.52e52 + 7.82e52i)T^{2} \) |
| 71 | \( 1 + 8.59e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (1.64e26 + 2.84e26i)T + (-5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-1.97e27 + 3.41e27i)T + (-5.37e54 - 9.30e54i)T^{2} \) |
| 83 | \( 1 - 9.02e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (7.89e27 - 1.36e28i)T + (-1.70e56 - 2.95e56i)T^{2} \) |
| 97 | \( 1 + 6.45e27T + 4.13e57T^{2} \) |
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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
−12.05802625924501542056114635784, −10.55882776941194414039929436418, −9.524415454772951129969118232486, −8.877597625670615524400208135264, −6.12197178947927952968856237152, −5.00800333720561358314073305775, −4.03838319337350717318735593458, −2.99100023068095886391286467691, −1.70367112616362247661041904896, −0.961192142346706071136821228198,
0.988277390772236875686071853254, 3.18240630624302039104483322739, 3.56696251068666642766882649872, 5.72867741758679130652727208876, 6.32246209133845895609831056054, 7.12072762703658956713820732864, 8.340028864370494378762736853014, 9.862060554400222093452146450048, 11.64894147277402172041564965594, 13.50358801490784263303055618734
