L(s) = 1 | + (−1.36 + 0.356i)2-s + i·3-s + (1.74 − 0.976i)4-s + (−1.65 − 1.50i)5-s + (−0.356 − 1.36i)6-s + (−2.07 − 1.64i)7-s + (−2.03 + 1.95i)8-s − 9-s + (2.79 + 1.47i)10-s + 4.97i·11-s + (0.976 + 1.74i)12-s + 5.35·13-s + (3.42 + 1.50i)14-s + (1.50 − 1.65i)15-s + (2.09 − 3.40i)16-s + 3.34·17-s + ⋯ |
L(s) = 1 | + (−0.967 + 0.252i)2-s + 0.577i·3-s + (0.872 − 0.488i)4-s + (−0.738 − 0.673i)5-s + (−0.145 − 0.558i)6-s + (−0.783 − 0.621i)7-s + (−0.721 + 0.692i)8-s − 0.333·9-s + (0.885 + 0.465i)10-s + 1.49i·11-s + (0.282 + 0.503i)12-s + 1.48·13-s + (0.915 + 0.403i)14-s + (0.388 − 0.426i)15-s + (0.522 − 0.852i)16-s + 0.810·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.720864 + 0.251513i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.720864 + 0.251513i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36 - 0.356i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.65 + 1.50i)T \) |
| 7 | \( 1 + (2.07 + 1.64i)T \) |
good | 11 | \( 1 - 4.97iT - 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 - 2.12T + 19T^{2} \) |
| 23 | \( 1 - 6.71T + 23T^{2} \) |
| 29 | \( 1 + 1.42T + 29T^{2} \) |
| 31 | \( 1 - 5.76T + 31T^{2} \) |
| 37 | \( 1 + 0.864iT - 37T^{2} \) |
| 41 | \( 1 + 1.68iT - 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 + 8.84iT - 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + 4.51T + 59T^{2} \) |
| 61 | \( 1 - 3.54iT - 61T^{2} \) |
| 67 | \( 1 - 0.310T + 67T^{2} \) |
| 71 | \( 1 + 4.74iT - 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 9.38iT - 79T^{2} \) |
| 83 | \( 1 - 6.05iT - 83T^{2} \) |
| 89 | \( 1 - 3.30iT - 89T^{2} \) |
| 97 | \( 1 + 5.32T + 97T^{2} \) |
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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
−11.02818278511614396616484412224, −10.23032421032100393489002694161, −9.431775202083370513219054868253, −8.722427984650573371592565439009, −7.65092376082095828638929466008, −6.92454534552787964624968400479, −5.65667831037651607830085330675, −4.39788230033386228204038306689, −3.22630402632818029370566493880, −1.08908305166154067780334342137,
0.905777387074236184900676859906, 2.97235421496664861957423817164, 3.41421365520667680837765743538, 5.92271206028089509628119644318, 6.48406589137412262138980232850, 7.58053402081318298134239959399, 8.427626625364800598449802608195, 9.057375778709161852658900012300, 10.30929045490243459118054360290, 11.24106274809444076368381444038