L(s) = 1 | + (0.707 + 0.707i)5-s − 1.69i·7-s + (−1.72 − 1.72i)11-s + (0.217 − 0.217i)13-s − 3.27·17-s + (−1.73 + 1.73i)19-s + 2.93i·23-s + 1.00i·25-s + (−6.42 + 6.42i)29-s − 7.75·31-s + (1.20 − 1.20i)35-s + (4.70 + 4.70i)37-s + 4.36i·41-s + (4.30 + 4.30i)43-s + 0.568·47-s + ⋯ |
L(s) = 1 | + (0.316 + 0.316i)5-s − 0.642i·7-s + (−0.521 − 0.521i)11-s + (0.0602 − 0.0602i)13-s − 0.795·17-s + (−0.397 + 0.397i)19-s + 0.611i·23-s + 0.200i·25-s + (−1.19 + 1.19i)29-s − 1.39·31-s + (0.203 − 0.203i)35-s + (0.772 + 0.772i)37-s + 0.681i·41-s + (0.656 + 0.656i)43-s + 0.0829·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8325967997\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8325967997\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + 1.69iT - 7T^{2} \) |
| 11 | \( 1 + (1.72 + 1.72i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.217 + 0.217i)T - 13iT^{2} \) |
| 17 | \( 1 + 3.27T + 17T^{2} \) |
| 19 | \( 1 + (1.73 - 1.73i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.93iT - 23T^{2} \) |
| 29 | \( 1 + (6.42 - 6.42i)T - 29iT^{2} \) |
| 31 | \( 1 + 7.75T + 31T^{2} \) |
| 37 | \( 1 + (-4.70 - 4.70i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.36iT - 41T^{2} \) |
| 43 | \( 1 + (-4.30 - 4.30i)T + 43iT^{2} \) |
| 47 | \( 1 - 0.568T + 47T^{2} \) |
| 53 | \( 1 + (-0.749 - 0.749i)T + 53iT^{2} \) |
| 59 | \( 1 + (-2.21 - 2.21i)T + 59iT^{2} \) |
| 61 | \( 1 + (-7.39 + 7.39i)T - 61iT^{2} \) |
| 67 | \( 1 + (9.00 - 9.00i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.32iT - 71T^{2} \) |
| 73 | \( 1 + 0.285iT - 73T^{2} \) |
| 79 | \( 1 - 2.59T + 79T^{2} \) |
| 83 | \( 1 + (10.7 - 10.7i)T - 83iT^{2} \) |
| 89 | \( 1 - 10.1iT - 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−9.059116700500186496596052643167, −8.231347076918941182327180774728, −7.44586347826413106790648176300, −6.83942075212165831094827991780, −5.92309043698836551473121722533, −5.28472687387283908854575417278, −4.21028784497434313293237211136, −3.44852988774446617952506443748, −2.45015196516421639823574856235, −1.32419331628571450898823855802,
0.25454877451884064930347777072, 2.00270599861529440902507045028, 2.46668326244713309435348127712, 3.86629874040993334380173068077, 4.62280062036637155963427812343, 5.54421822677212787445419988673, 6.06036732249089952993059461667, 7.11238133892619608961924167333, 7.70824550873879230626463246336, 8.831526349675441120462800492069