L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.382 − 0.662i)5-s + 0.999i·8-s + (0.662 + 0.382i)10-s + 1.84i·13-s + (−0.5 − 0.866i)16-s + (−0.923 + 1.60i)17-s − 0.765·20-s + (0.207 − 0.358i)25-s + (−0.923 − 1.60i)26-s + (0.866 + 0.499i)32-s − 1.84i·34-s + (0.707 + 1.22i)37-s + (0.662 − 0.382i)40-s + 1.84·41-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.382 − 0.662i)5-s + 0.999i·8-s + (0.662 + 0.382i)10-s + 1.84i·13-s + (−0.5 − 0.866i)16-s + (−0.923 + 1.60i)17-s − 0.765·20-s + (0.207 − 0.358i)25-s + (−0.923 − 1.60i)26-s + (0.866 + 0.499i)32-s − 1.84i·34-s + (0.707 + 1.22i)37-s + (0.662 − 0.382i)40-s + 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6193420538\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6193420538\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.84iT - T^{2} \) |
| 17 | \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - 1.84T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.662 + 0.382i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + 0.765iT - T^{2} \) |
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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
−9.345541420343739145173080212726, −8.828626660037607492848746932698, −8.259758126805594419582186473612, −7.36997402768494892077205314187, −6.50787531646584362198634300962, −5.96529943918581877770064490754, −4.64361388691766395828997011643, −4.14134741111311213421316270759, −2.36284917591724346278549527624, −1.34426446403553241771102251631,
0.67258309858569759672237979264, 2.47028648725346020480655871121, 3.02678657701290230155561098878, 4.04623941326862038703188772671, 5.27683123465943766920045376520, 6.35166698732216635187087763693, 7.43087924604296477024190561271, 7.55758733087786431539534897867, 8.647418473299966477314071461403, 9.354733112996554974344086078757