L(s) = 1 | + i·2-s − 4-s − i·8-s − 2i·11-s + 16-s + 2·22-s − 2i·23-s − 25-s + i·32-s + 2·37-s + 2i·44-s + 2·46-s − i·50-s − 64-s + 2i·71-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s − i·8-s − 2i·11-s + 16-s + 2·22-s − 2i·23-s − 25-s + i·32-s + 2·37-s + 2i·44-s + 2·46-s − i·50-s − 64-s + 2i·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9161266869\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9161266869\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + 2iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 2iT - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 2T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 2iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.246431659450565247006921931598, −8.405083282613219135238751943892, −8.136044192715390128572271088127, −7.04563518481578072468631546470, −6.10961685892133376495046241716, −5.81092166993369817941610123598, −4.64474357843096484590921649091, −3.81010106382271513261700290318, −2.75162345285640079406163651692, −0.74598289977922072572841224702,
1.54863206611834761541176199307, 2.36410989565923920772025707143, 3.58397151224920566203758156222, 4.40399428588893855454513683357, 5.14495973646711114181274785782, 6.14094826546581014578261410632, 7.46584590131062865531149361679, 7.79851068674101803897170280456, 9.087226457401992645957959057159, 9.677534016422554380889696489099