L(s) = 1 | + (−1.33 − 0.475i)2-s + i·3-s + (1.54 + 1.26i)4-s + (−2.22 − 0.260i)5-s + (0.475 − 1.33i)6-s + (1.17 + 2.36i)7-s + (−1.45 − 2.42i)8-s − 9-s + (2.83 + 1.40i)10-s + 0.365i·11-s + (−1.26 + 1.54i)12-s − 1.72·13-s + (−0.442 − 3.71i)14-s + (0.260 − 2.22i)15-s + (0.790 + 3.92i)16-s + 0.146·17-s + ⋯ |
L(s) = 1 | + (−0.941 − 0.336i)2-s + 0.577i·3-s + (0.773 + 0.633i)4-s + (−0.993 − 0.116i)5-s + (0.194 − 0.543i)6-s + (0.445 + 0.895i)7-s + (−0.515 − 0.856i)8-s − 0.333·9-s + (0.896 + 0.443i)10-s + 0.110i·11-s + (−0.365 + 0.446i)12-s − 0.479·13-s + (−0.118 − 0.992i)14-s + (0.0672 − 0.573i)15-s + (0.197 + 0.980i)16-s + 0.0356·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0475035 + 0.275841i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0475035 + 0.275841i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.33 + 0.475i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.22 + 0.260i)T \) |
| 7 | \( 1 + (-1.17 - 2.36i)T \) |
good | 11 | \( 1 - 0.365iT - 11T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 - 0.146T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 + 3.44T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 + 6.07T + 31T^{2} \) |
| 37 | \( 1 + 7.52iT - 37T^{2} \) |
| 41 | \( 1 - 7.16iT - 41T^{2} \) |
| 43 | \( 1 + 8.65T + 43T^{2} \) |
| 47 | \( 1 - 3.83iT - 47T^{2} \) |
| 53 | \( 1 - 11.5iT - 53T^{2} \) |
| 59 | \( 1 + 6.18T + 59T^{2} \) |
| 61 | \( 1 + 0.348iT - 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 7.78iT - 71T^{2} \) |
| 73 | \( 1 - 4.55T + 73T^{2} \) |
| 79 | \( 1 - 14.2iT - 79T^{2} \) |
| 83 | \( 1 + 3.53iT - 83T^{2} \) |
| 89 | \( 1 + 13.6iT - 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−11.33102030229761882644292421419, −10.83736117129941183519068531511, −9.674705405740123970345526942274, −8.898651644115154200442223010783, −8.155633270773711056408673935546, −7.34607744305423495039521417718, −5.99369378508136046824303566183, −4.61394983730811734389533914927, −3.48356672013708517903815949699, −2.12218218000962105047812631026,
0.23025606138694802409148338509, 1.92660704018117432550605964350, 3.66942419214430160775706166219, 5.09671698137200227527273976571, 6.54339722375604485228260309834, 7.24413217872672893048236933301, 7.989371473541828781258233128295, 8.631696528176238156207644039897, 9.934848703820461650559170552641, 10.82411010662492596636352695414