L(s) = 1 | + 2.23i·2-s − 3.00·4-s − i·7-s − 2.23i·8-s + 2.47·11-s − 4.47i·13-s + 2.23·14-s − 0.999·16-s − 2i·17-s − 6.47·19-s + 5.52i·22-s − 4i·23-s + 10.0·26-s + 3.00i·28-s − 2·29-s + ⋯ |
L(s) = 1 | + 1.58i·2-s − 1.50·4-s − 0.377i·7-s − 0.790i·8-s + 0.745·11-s − 1.24i·13-s + 0.597·14-s − 0.249·16-s − 0.485i·17-s − 1.48·19-s + 1.17i·22-s − 0.834i·23-s + 1.96·26-s + 0.566i·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.264785488\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.264785488\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 2 | \( 1 - 2.23iT - 2T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 10.9iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.94iT - 43T^{2} \) |
| 47 | \( 1 + 4.94iT - 47T^{2} \) |
| 53 | \( 1 - 12.4iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 3.52iT - 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 + 0.944iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 0.472iT - 97T^{2} \) |
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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
−9.032250873622069343136730375643, −8.564321862022177825130233254823, −7.73727690617323702097185833999, −7.06161758703418595512119930638, −6.30026191335359124632248968138, −5.66767158298693579228743521215, −4.65841611519778438721181008705, −3.93818049270252831430957372544, −2.46766060647145738656799321563, −0.52542446725463026152725304473,
1.32632172643212038205313858749, 2.14114308615981878430954931506, 3.18642491838913563963657999735, 4.17970849162196801935524924637, 4.69514735753630666949375857691, 6.20308931677202613031965305342, 6.74838916203258435424600027745, 8.200796551138726344372244733816, 8.833196703049690641057940127817, 9.615790739794429305258558438024