L(s) = 1 | + 2i·2-s + 9.84·3-s − 4·4-s + 14.4i·5-s + 19.6i·6-s − 22.1·7-s − 8i·8-s + 69.8·9-s − 28.8·10-s + 21.0·11-s − 39.3·12-s − 77.2i·13-s − 44.3i·14-s + 141. i·15-s + 16·16-s + 73.4i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.89·3-s − 0.5·4-s + 1.28i·5-s + 1.33i·6-s − 1.19·7-s − 0.353i·8-s + 2.58·9-s − 0.911·10-s + 0.576·11-s − 0.947·12-s − 1.64i·13-s − 0.847i·14-s + 2.44i·15-s + 0.250·16-s + 1.04i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.78996 + 1.44096i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78996 + 1.44096i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 37 | \( 1 + (219. + 48.0i)T \) |
good | 3 | \( 1 - 9.84T + 27T^{2} \) |
| 5 | \( 1 - 14.4iT - 125T^{2} \) |
| 7 | \( 1 + 22.1T + 343T^{2} \) |
| 11 | \( 1 - 21.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 77.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 73.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 68.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 25.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 268. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 86.5iT - 2.97e4T^{2} \) |
| 41 | \( 1 - 5.19T + 6.89e4T^{2} \) |
| 43 | \( 1 - 125. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 181.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 63.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 207. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 299. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 142.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 501.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 200.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.07e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 154. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 829. iT - 9.12e5T^{2} \) |
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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
−14.50224484772684318414084524417, −13.50678106764594623689289253135, −12.78962938711871678284063575000, −10.39776607205581100140950743979, −9.634689055225641360145818752463, −8.415606221179064082673947934740, −7.37052451512932012957728949097, −6.37773379501088365090330668500, −3.70312467222885868317765993402, −2.84243861556447914107317339178,
1.65766942906267841588348876238, 3.33445681914912380303200567535, 4.46526820581421215375895934330, 7.03076724246144656422681832384, 8.694414655761520499350492428262, 9.149739208433809696435909572424, 9.903781535194843473758848170952, 12.06489457474869742211352130316, 12.88014646361237273160923861618, 13.74495382750959728075767661838