L(s) = 1 | + i·2-s + 3-s − 4-s + i·5-s + i·6-s + 0.561i·7-s − i·8-s + 9-s − 10-s − 4.12i·11-s − 12-s − 0.561·14-s + i·15-s + 16-s + 3.12·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s + 0.212i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 1.24i·11-s − 0.288·12-s − 0.150·14-s + 0.258i·15-s + 0.250·16-s + 0.757·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.257972943\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.257972943\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 0.561iT - 7T^{2} \) |
| 11 | \( 1 + 4.12iT - 11T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 - 0.561iT - 19T^{2} \) |
| 23 | \( 1 + 4.68T + 23T^{2} \) |
| 29 | \( 1 - 2.43T + 29T^{2} \) |
| 31 | \( 1 + 6.68iT - 31T^{2} \) |
| 37 | \( 1 - 4.12iT - 37T^{2} \) |
| 41 | \( 1 - 12.2iT - 41T^{2} \) |
| 43 | \( 1 + 0.438T + 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 + 6.43iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 2.24iT - 67T^{2} \) |
| 71 | \( 1 + 13.1iT - 71T^{2} \) |
| 73 | \( 1 + 9.36iT - 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 7.12iT - 83T^{2} \) |
| 89 | \( 1 - 18.8iT - 89T^{2} \) |
| 97 | \( 1 + 7.12iT - 97T^{2} \) |
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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
−8.067630158265301850178717815434, −7.85320880767225055555942526253, −6.82589819519033580802173460927, −6.15367656920580797926134706635, −5.62477600849480791856739472976, −4.64486926332330398739263864057, −3.69151670939179863190118288764, −3.14132744091981522961222612446, −2.10028671165813475738426091775, −0.71350430484172007267513298975,
0.917147537413375101565760499812, 1.91360179542948529244130536116, 2.61558717514862029259951904081, 3.73535548125441095344065664150, 4.21231389744182722648495147868, 5.07286117538145418600262258376, 5.79201431739674050081430378896, 7.03546270235448334937820289081, 7.45207072969895358055660617399, 8.368119895551277233981933935106