L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (0.923 − 1.60i)5-s − 0.999i·8-s + (−1.60 + 0.923i)10-s − 0.765i·13-s + (−0.5 + 0.866i)16-s + (−0.382 − 0.662i)17-s + 1.84·20-s + (−1.20 − 2.09i)25-s + (−0.382 + 0.662i)26-s + (0.866 − 0.499i)32-s + 0.765i·34-s + (−0.707 + 1.22i)37-s + (−1.60 − 0.923i)40-s + 0.765·41-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (0.923 − 1.60i)5-s − 0.999i·8-s + (−1.60 + 0.923i)10-s − 0.765i·13-s + (−0.5 + 0.866i)16-s + (−0.382 − 0.662i)17-s + 1.84·20-s + (−1.20 − 2.09i)25-s + (−0.382 + 0.662i)26-s + (0.866 − 0.499i)32-s + 0.765i·34-s + (−0.707 + 1.22i)37-s + (−1.60 − 0.923i)40-s + 0.765·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8429109180\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8429109180\) |
\(L(1)\) |
|
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 | \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + 0.765iT - T^{2} \) |
| 17 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 - 0.765T + T^{2} \) |
| 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 | \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 1.84iT - T^{2} \) |
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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
−9.368239203677408910131623686353, −8.529207501125716095487817596266, −8.093967962894371485037529882259, −7.00635064601897715304066386048, −5.99920618460051704089451217510, −5.13368044988761498334716832833, −4.28379365450596330561295762791, −2.94960379225934915968711266153, −1.88203065044602014363585955014, −0.857297185712590379902205919648,
1.81146344584377180028351987538, 2.49490116475963109977274232561, 3.74500194172283528032333783069, 5.22258793155786761799572264624, 6.09879249611845345304274868716, 6.64247607413064921161971749778, 7.20667561940764863530675549710, 8.114419957337817731373389286175, 9.112675925622376113226001953849, 9.690907288258332643245851916780