| L(s) = 1 | − 1.56i·3-s + 3.12i·7-s + 0.561·9-s − 4·11-s + 3.56i·13-s − 5.12i·17-s − 4·19-s + 4.87·21-s + i·23-s − 5.56i·27-s + 4.43·29-s + 5.56·31-s + 6.24i·33-s − 1.12i·37-s + 5.56·39-s + ⋯ |
| L(s) = 1 | − 0.901i·3-s + 1.18i·7-s + 0.187·9-s − 1.20·11-s + 0.987i·13-s − 1.24i·17-s − 0.917·19-s + 1.06·21-s + 0.208i·23-s − 1.07i·27-s + 0.824·29-s + 0.998·31-s + 1.08i·33-s − 0.184i·37-s + 0.890·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6197114129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6197114129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.56iT - 3T^{2} \) |
| 7 | \( 1 - 3.12iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 3.56iT - 13T^{2} \) |
| 17 | \( 1 + 5.12iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 + 1.12iT - 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 + 0.876iT - 43T^{2} \) |
| 47 | \( 1 + 8.68iT - 47T^{2} \) |
| 53 | \( 1 - 12.2iT - 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 - 10.2iT - 67T^{2} \) |
| 71 | \( 1 + 8.68T + 71T^{2} \) |
| 73 | \( 1 - 12.4iT - 73T^{2} \) |
| 79 | \( 1 + 6.24T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 0.246iT - 97T^{2} \) |
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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
−8.499819388909869079787569431502, −7.81243025857132081408312301153, −7.04938757430197810085809961673, −6.51803645764681538518849035885, −5.68833009965830259137738192201, −4.95863565939844746628080249133, −4.17034983856584473864492827232, −2.62283112932655061395816355007, −2.47411355451289419134388882183, −1.29158433628777056655537454814,
0.16962741683272865265108465500, 1.50736981293879733787855733832, 2.84627001412334676228937280838, 3.58350835055198882615681783056, 4.46556837653542659029556557744, 4.81190604587418787663022566939, 5.85967092972510619297879293103, 6.59554944220183163588087601666, 7.51799880822784631579552233509, 8.099976068702529181755186711574
