L(s) = 1 | + (0.5 + 0.866i)3-s + (1 − 1.73i)5-s + (2.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + 13-s + 1.99·15-s + (0.5 − 0.866i)19-s + (2 + 1.73i)21-s + (0.500 + 0.866i)25-s − 0.999·27-s + 4·29-s + (4.5 + 7.79i)31-s + (0.999 − 1.73i)33-s + (1.00 − 5.19i)35-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.447 − 0.774i)5-s + (0.944 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + 0.277·13-s + 0.516·15-s + (0.114 − 0.198i)19-s + (0.436 + 0.377i)21-s + (0.100 + 0.173i)25-s − 0.192·27-s + 0.742·29-s + (0.808 + 1.39i)31-s + (0.174 − 0.301i)33-s + (0.169 − 0.878i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.65947 - 0.105310i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.65947 - 0.105310i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (-4.5 - 7.79i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (8 - 13.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−11.40600450184990641011993417741, −10.58122589564776834266931025629, −9.683132609076294109829606803332, −8.575184208198096266151663157800, −8.163797806184969046578832758502, −6.69096001391946685916152534534, −5.26448461159680183834924613663, −4.69813009178533278507597604460, −3.22664450895118351570326821253, −1.47431717086110302589789240265,
1.79967279993630797846841592914, 2.91328961523927284058033257501, 4.56022737392867558812610682089, 5.82661766187358698849374095707, 6.80391003450114410908091692358, 7.80582835615406115758091385680, 8.596179846730826081126212669275, 9.807703268306811534641644660209, 10.64833204944929744237983947565, 11.61939431803824956055912989940