| L(s) = 1 | − i·3-s + 2·9-s − 3·11-s − 2i·13-s + 3i·17-s + 19-s − 3i·23-s − 5i·27-s + 6·29-s − 7·31-s + 3i·33-s − i·37-s − 2·39-s + 6·41-s + 4i·43-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.666·9-s − 0.904·11-s − 0.554i·13-s + 0.727i·17-s + 0.229·19-s − 0.625i·23-s − 0.962i·27-s + 1.11·29-s − 1.25·31-s + 0.522i·33-s − 0.164i·37-s − 0.320·39-s + 0.937·41-s + 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.474259539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.474259539\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 3iT - 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−7.899098379217699010317305877320, −7.43020455411249527898997160543, −6.61555423558670207069804360856, −5.94740582153883399883830238399, −5.11339649328324959295436506992, −4.35460785784219168272298818754, −3.39088471727361289276088172471, −2.46648521823650946641573969629, −1.58649462759805609457398512738, −0.42397823154395648778831092447,
1.15531810031516514765978002839, 2.32034610397147857820249294163, 3.21682772737196117589182888027, 4.09989821521062368767596242666, 4.80329347817667206268399518361, 5.40501748438879576785780054901, 6.31459565782867269578264729617, 7.24311360292982166726093300436, 7.61399396145689947455611681360, 8.559615371673380235525748837859
