L(s) = 1 | − 2.41i·2-s − i·3-s − 3.82·4-s − 2.41·6-s − 2.82i·7-s + 4.41i·8-s − 9-s − 2·11-s + 3.82i·12-s + i·13-s − 6.82·14-s + 2.99·16-s − 3.65i·17-s + 2.41i·18-s − 2.82·19-s + ⋯ |
L(s) = 1 | − 1.70i·2-s − 0.577i·3-s − 1.91·4-s − 0.985·6-s − 1.06i·7-s + 1.56i·8-s − 0.333·9-s − 0.603·11-s + 1.10i·12-s + 0.277i·13-s − 1.82·14-s + 0.749·16-s − 0.886i·17-s + 0.569i·18-s − 0.648·19-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
\(L(1)\) |
\(\approx\) |
\(0.543869 + 0.336129i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.543869 + 0.336129i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.41iT - 2T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 3.65iT - 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 3.65iT - 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 9.65iT - 43T^{2} \) |
| 47 | \( 1 + 0.343iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 1.17iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 11.6iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 7.65iT - 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 + 7.65iT - 97T^{2} \) |
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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
−9.548010188425038212327043437520, −8.822577884279626949511310629623, −7.66304732669546696913337246137, −7.01345150121244380997268399794, −5.59148833463464964694904636914, −4.49224849754064866300322219559, −3.64130468942501621101632294348, −2.60186256966660128009654903175, −1.54404130552665583382355219566, −0.29447920915469462073745020250,
2.48329095870409955935573450255, 3.98509939943295872239769294276, 4.91034121126452710860501708403, 5.76812870874958251909192061529, 6.18050745953012928292312499152, 7.38328318365397481590527685071, 8.184213187641571372812647508007, 8.802602219291349782385387471679, 9.444791562545566582866647218748, 10.49370504414654693588041742015