L(s) = 1 | + (2.68 − 2.68i)5-s + 2.15i·7-s + (1.79 − 1.79i)11-s + (1.38 + 1.38i)13-s + 0.224·17-s + (−0.158 − 0.158i)19-s − 2.82i·23-s − 9.42i·25-s + (1.85 + 1.85i)29-s − 1.84·31-s + (5.79 + 5.79i)35-s + (−3.66 + 3.66i)37-s − 5.88i·41-s + (7.75 − 7.75i)43-s − 2.82·47-s + ⋯ |
L(s) = 1 | + (1.20 − 1.20i)5-s + 0.816i·7-s + (0.542 − 0.542i)11-s + (0.383 + 0.383i)13-s + 0.0545·17-s + (−0.0364 − 0.0364i)19-s − 0.589i·23-s − 1.88i·25-s + (0.344 + 0.344i)29-s − 0.330·31-s + (0.980 + 0.980i)35-s + (−0.603 + 0.603i)37-s − 0.918i·41-s + (1.18 − 1.18i)43-s − 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.77583 - 0.491166i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77583 - 0.491166i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.68 + 2.68i)T - 5iT^{2} \) |
| 7 | \( 1 - 2.15iT - 7T^{2} \) |
| 11 | \( 1 + (-1.79 + 1.79i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.38 - 1.38i)T + 13iT^{2} \) |
| 17 | \( 1 - 0.224T + 17T^{2} \) |
| 19 | \( 1 + (0.158 + 0.158i)T + 19iT^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (-1.85 - 1.85i)T + 29iT^{2} \) |
| 31 | \( 1 + 1.84T + 31T^{2} \) |
| 37 | \( 1 + (3.66 - 3.66i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.88iT - 41T^{2} \) |
| 43 | \( 1 + (-7.75 + 7.75i)T - 43iT^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + (7.51 - 7.51i)T - 53iT^{2} \) |
| 59 | \( 1 + (-4 + 4i)T - 59iT^{2} \) |
| 61 | \( 1 + (-5.98 - 5.98i)T + 61iT^{2} \) |
| 67 | \( 1 + (-10.4 - 10.4i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.31iT - 71T^{2} \) |
| 73 | \( 1 - 5.97iT - 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + (10.1 + 10.1i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.42iT - 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−10.52766243624315980537546620797, −9.575651569658471797224920473370, −8.812172047346572737440575556180, −8.504684374414577117546673401203, −6.84590380981995130880442894990, −5.85013110827075233514784414324, −5.31286622834080881661701657293, −4.10993928216977626968194442034, −2.47111821657142522495418113066, −1.27433624356110501397857124153,
1.60373925178502971529313638343, 2.90112530336452409137645105546, 4.01835546503428154920815679906, 5.42987564563612803745623732534, 6.40917011160616741632578321338, 7.01020406384434705920221386370, 7.995357825474630104077578471236, 9.427558186801703624502929612273, 9.902463128647903869554604021566, 10.75504517686956475488330532863