| L(s) = 1 | + (0.777 − 1.54i)3-s − 3.40i·5-s + (−1.79 − 2.40i)9-s − 3.94·13-s + (−5.26 − 2.64i)15-s + 6.09i·17-s + 4.38i·19-s − 8.33·23-s − 6.58·25-s + (−5.11 + 0.901i)27-s − 3.80i·29-s + 4.89i·31-s − 7.58·37-s + (−3.06 + 6.10i)39-s − 0.710i·41-s + ⋯ |
| L(s) = 1 | + (0.448 − 0.893i)3-s − 1.52i·5-s + (−0.597 − 0.802i)9-s − 1.09·13-s + (−1.36 − 0.683i)15-s + 1.47i·17-s + 1.00i·19-s − 1.73·23-s − 1.31·25-s + (−0.984 + 0.173i)27-s − 0.707i·29-s + 0.879i·31-s − 1.24·37-s + (−0.491 + 0.978i)39-s − 0.110i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5207120664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5207120664\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.777 + 1.54i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.40iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 3.94T + 13T^{2} \) |
| 17 | \( 1 - 6.09iT - 17T^{2} \) |
| 19 | \( 1 - 4.38iT - 19T^{2} \) |
| 23 | \( 1 + 8.33T + 23T^{2} \) |
| 29 | \( 1 + 3.80iT - 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 + 7.58T + 37T^{2} \) |
| 41 | \( 1 + 0.710iT - 41T^{2} \) |
| 43 | \( 1 + 9.66iT - 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 1.00iT - 53T^{2} \) |
| 59 | \( 1 - 4.66T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 6.59T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 6.09iT - 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−8.442230697199038271755223086211, −7.893581182925081964781382603258, −7.11291469797943534528695418383, −5.98055828702926943163719675752, −5.52224515605127385409334159509, −4.35532833846334047721738631606, −3.64211423403980983020557840165, −2.14398339864063607018332123047, −1.52851179465473193697867683737, −0.15224976525002371117795906877,
2.39679489895991195732619991051, 2.74109051030275110909486353074, 3.72764343632805869434828180556, 4.63774721367211618094612883528, 5.46892108224248833440709893109, 6.48440559600941383218942631187, 7.33644558428990823710377363360, 7.74573970896932165127351812661, 8.946192309945739296169644156743, 9.569899601221041867992515243487
