L(s) = 1 | + (1.38 + 1.04i)3-s + (−0.707 + 0.707i)7-s + (0.829 + 2.88i)9-s + 1.21i·11-s + (1.31 + 1.31i)13-s + (−4.79 − 4.79i)17-s + 3.66i·19-s + (−1.71 + 0.241i)21-s + (−6.15 + 6.15i)23-s + (−1.85 + 4.85i)27-s + 6.65·29-s + 0.677·31-s + (−1.26 + 1.67i)33-s + (−5.16 + 5.16i)37-s + (0.450 + 3.19i)39-s + ⋯ |
L(s) = 1 | + (0.798 + 0.601i)3-s + (−0.267 + 0.267i)7-s + (0.276 + 0.961i)9-s + 0.365i·11-s + (0.365 + 0.365i)13-s + (−1.16 − 1.16i)17-s + 0.841i·19-s + (−0.374 + 0.0527i)21-s + (−1.28 + 1.28i)23-s + (−0.357 + 0.934i)27-s + 1.23·29-s + 0.121·31-s + (−0.219 + 0.291i)33-s + (−0.849 + 0.849i)37-s + (0.0721 + 0.512i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.709568178\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.709568178\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.38 - 1.04i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 - 1.21iT - 11T^{2} \) |
| 13 | \( 1 + (-1.31 - 1.31i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.79 + 4.79i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.66iT - 19T^{2} \) |
| 23 | \( 1 + (6.15 - 6.15i)T - 23iT^{2} \) |
| 29 | \( 1 - 6.65T + 29T^{2} \) |
| 31 | \( 1 - 0.677T + 31T^{2} \) |
| 37 | \( 1 + (5.16 - 5.16i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.65iT - 41T^{2} \) |
| 43 | \( 1 + (-2.35 - 2.35i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.91 + 1.91i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.64 + 8.64i)T - 53iT^{2} \) |
| 59 | \( 1 + 6.62T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 + (4.94 - 4.94i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.87iT - 71T^{2} \) |
| 73 | \( 1 + (-7.93 - 7.93i)T + 73iT^{2} \) |
| 79 | \( 1 - 10.6iT - 79T^{2} \) |
| 83 | \( 1 + (-9.69 + 9.69i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.84T + 89T^{2} \) |
| 97 | \( 1 + (3.90 - 3.90i)T - 97iT^{2} \) |
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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
−9.474783979018641566925847914977, −8.631442890619107422736908678240, −8.028643702260588534556745585097, −7.12696792965857764291459081129, −6.29728548732024540161057066691, −5.22483494463727040675792104417, −4.43507158543269067926633046770, −3.61370552860995262980043915333, −2.67390321531137622510953860993, −1.71417429684275932830454335850,
0.51211555611829742544784072490, 1.91761865846680198987108243702, 2.79681132005574841706842917434, 3.81111486727529130374654126915, 4.52761806579064217096657393938, 6.00844606553332204191571476561, 6.47135240864330419104095745398, 7.29788064381197339875092531531, 8.212574288387996591055038703549, 8.679869874295704085616815330192