L(s) = 1 | + (−0.707 − 0.707i)5-s − 1.44·7-s + (−1.89 + 1.89i)11-s + (0.905 + 0.905i)13-s − 6.42i·17-s + (2.02 − 2.02i)19-s + 4.25i·23-s + 1.00i·25-s + (2.71 − 2.71i)29-s + 4.19i·31-s + (1.02 + 1.02i)35-s + (0.0486 − 0.0486i)37-s − 8.11·41-s + (3.17 + 3.17i)43-s − 2.47·47-s + ⋯ |
L(s) = 1 | + (−0.316 − 0.316i)5-s − 0.546·7-s + (−0.571 + 0.571i)11-s + (0.251 + 0.251i)13-s − 1.55i·17-s + (0.464 − 0.464i)19-s + 0.887i·23-s + 0.200i·25-s + (0.504 − 0.504i)29-s + 0.752i·31-s + (0.172 + 0.172i)35-s + (0.00800 − 0.00800i)37-s − 1.26·41-s + (0.484 + 0.484i)43-s − 0.361·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6090700704\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6090700704\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 + (1.89 - 1.89i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.905 - 0.905i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.42iT - 17T^{2} \) |
| 19 | \( 1 + (-2.02 + 2.02i)T - 19iT^{2} \) |
| 23 | \( 1 - 4.25iT - 23T^{2} \) |
| 29 | \( 1 + (-2.71 + 2.71i)T - 29iT^{2} \) |
| 31 | \( 1 - 4.19iT - 31T^{2} \) |
| 37 | \( 1 + (-0.0486 + 0.0486i)T - 37iT^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 + (-3.17 - 3.17i)T + 43iT^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 + (-1.67 - 1.67i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.69 - 4.69i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.69 - 2.69i)T + 61iT^{2} \) |
| 67 | \( 1 + (8.13 - 8.13i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 - 5.23iT - 73T^{2} \) |
| 79 | \( 1 - 6.59iT - 79T^{2} \) |
| 83 | \( 1 + (12.8 + 12.8i)T + 83iT^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−9.035453590379403306415739069712, −8.320003697491441921005998277224, −7.28965221020184268124786066472, −7.05022468003292719198553780990, −5.90133183684393474891271059962, −5.07325540836975173606260829283, −4.44123411077673097218485527263, −3.31319648113533700533930361809, −2.59010344404440944669693853788, −1.18997006931509862513204426146,
0.20599604457263724260159797710, 1.72880341358358565881487707608, 2.99755871817167416125802169798, 3.56509235825869998425655207898, 4.53261103254967635613048227544, 5.60222306908195620961402911611, 6.24144597642477095286992857630, 6.91956065671862085376329931368, 8.037784122086218194670170909279, 8.268587554948845337662032956272