| L(s) = 1 | + 2i·2-s + i·3-s − 2·4-s − 2·6-s + 3i·7-s − 9-s − 11-s − 2i·12-s + i·13-s − 6·14-s − 4·16-s − i·17-s − 2i·18-s + 2·19-s − 3·21-s − 2i·22-s + ⋯ |
| L(s) = 1 | + 1.41i·2-s + 0.577i·3-s − 4-s − 0.816·6-s + 1.13i·7-s − 0.333·9-s − 0.301·11-s − 0.577i·12-s + 0.277i·13-s − 1.60·14-s − 16-s − 0.242i·17-s − 0.471i·18-s + 0.458·19-s − 0.654·21-s − 0.426i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.603287 - 0.976139i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.603287 - 0.976139i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 - 2iT - 2T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 11iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 + 11iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 - 17iT - 97T^{2} \) |
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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
−10.31735317396824768809502703642, −9.422242296943292539054103396031, −8.735931152600397430769209815601, −8.113749870469077080885296620989, −7.11358636170508588123127764353, −6.29844590597326332851426982424, −5.32396479233902370931894440031, −5.02313578734529331417150860930, −3.55769438444294647887380150436, −2.25390446665196758907232158288,
0.52374012234064004903235829385, 1.64623953612891172505083644100, 2.81640120225307199592208901711, 3.74830190083547619016975775396, 4.67094723650139846474220326037, 5.98898085612776500076704849768, 7.09344020224328043004654971087, 7.70514096117492222865427021471, 8.850491829005862190419234371682, 9.706071431802248887959096682323
