| L(s) = 1 | + 3i·9-s + (5 + 5i)13-s + (5 − 5i)17-s + 4i·29-s + (−5 + 5i)37-s + 8·41-s − 7i·49-s + (−5 − 5i)53-s − 12·61-s + (−5 − 5i)73-s − 9·81-s − 16i·89-s + (−5 + 5i)97-s + 2·101-s + 6i·109-s + ⋯ |
| L(s) = 1 | + i·9-s + (1.38 + 1.38i)13-s + (1.21 − 1.21i)17-s + 0.742i·29-s + (−0.821 + 0.821i)37-s + 1.24·41-s − i·49-s + (−0.686 − 0.686i)53-s − 1.53·61-s + (−0.585 − 0.585i)73-s − 81-s − 1.69i·89-s + (−0.507 + 0.507i)97-s + 0.199·101-s + 0.574i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37935 + 0.391845i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37935 + 0.391845i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-5 - 5i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5 + 5i)T - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (5 + 5i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (5 + 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (5 - 5i)T - 97iT^{2} \) |
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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
−11.34389581752958628359760485054, −10.54545353312228005749820967794, −9.493423312652212097735151495875, −8.629770810801943550226109079514, −7.64187653849720920038089449727, −6.68928335851737744391590006756, −5.52846652566638912601909530456, −4.50293949880647670934962873573, −3.19380337442781207459228011090, −1.61431408663841428470958862339,
1.13266468726042336887303247194, 3.15925999426796568586280927185, 4.00464382900555206770790773095, 5.71084731742201172430389379938, 6.17528400162867302190580578900, 7.61950321962881613564710904487, 8.394194096742075503759627841014, 9.368931877555665881972871881716, 10.38665363389198613782678020822, 11.04472555806920005307368319708
