L(s) = 1 | + (0.290 + 0.290i)2-s + 3-s − 1.83i·4-s + (1.55 − 1.55i)5-s + (0.290 + 0.290i)6-s + (−0.786 + 0.786i)7-s + (1.11 − 1.11i)8-s + 9-s + 0.902·10-s + (−1.44 + 2.98i)11-s − 1.83i·12-s + (2.05 − 2.96i)13-s − 0.457·14-s + (1.55 − 1.55i)15-s − 3.01·16-s + 2.52·17-s + ⋯ |
L(s) = 1 | + (0.205 + 0.205i)2-s + 0.577·3-s − 0.915i·4-s + (0.694 − 0.694i)5-s + (0.118 + 0.118i)6-s + (−0.297 + 0.297i)7-s + (0.393 − 0.393i)8-s + 0.333·9-s + 0.285·10-s + (−0.434 + 0.900i)11-s − 0.528i·12-s + (0.568 − 0.822i)13-s − 0.122·14-s + (0.400 − 0.400i)15-s − 0.753·16-s + 0.613·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.86715 - 0.746012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86715 - 0.746012i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 + (1.44 - 2.98i)T \) |
| 13 | \( 1 + (-2.05 + 2.96i)T \) |
good | 2 | \( 1 + (-0.290 - 0.290i)T + 2iT^{2} \) |
| 5 | \( 1 + (-1.55 + 1.55i)T - 5iT^{2} \) |
| 7 | \( 1 + (0.786 - 0.786i)T - 7iT^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 + (1.03 + 1.03i)T + 19iT^{2} \) |
| 23 | \( 1 + 5.04iT - 23T^{2} \) |
| 29 | \( 1 + 0.703iT - 29T^{2} \) |
| 31 | \( 1 + (2.73 - 2.73i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.75 - 1.75i)T + 37iT^{2} \) |
| 41 | \( 1 + (-7.42 - 7.42i)T + 41iT^{2} \) |
| 43 | \( 1 + 9.78T + 43T^{2} \) |
| 47 | \( 1 + (-2.29 - 2.29i)T + 47iT^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + (2.75 + 2.75i)T + 59iT^{2} \) |
| 61 | \( 1 - 10.7iT - 61T^{2} \) |
| 67 | \( 1 + (9.79 - 9.79i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.26 - 2.26i)T - 71iT^{2} \) |
| 73 | \( 1 + (0.116 - 0.116i)T - 73iT^{2} \) |
| 79 | \( 1 + 15.3iT - 79T^{2} \) |
| 83 | \( 1 + (-4.15 - 4.15i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.00 + 2.00i)T + 89iT^{2} \) |
| 97 | \( 1 + (3.65 - 3.65i)T - 97iT^{2} \) |
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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
−10.71453126916364194873086024381, −10.04570059263907868482439803564, −9.334283147542893447963409300914, −8.475574343272906506382289783553, −7.29592581667873661205667841276, −6.14005441790301618207284770224, −5.35118365789433148279942952075, −4.40238007425670355475312770716, −2.68437783236168623806324385809, −1.33186036236133193649226946182,
2.09951610268075847427435124701, 3.23778201728696265363471961702, 3.96318278281593630688827968819, 5.60698065611421299343687478124, 6.72161076700478302239131141832, 7.61649545275641782196214881522, 8.536823184587154630562585291819, 9.424483126830909463806714847752, 10.45164771886389865397794990613, 11.23207185764022684883927919153