L(s) = 1 | + i·5-s + 2.44i·7-s − 1.01·11-s + 2.24·13-s + 4.89i·19-s − 8.36·23-s − 25-s + 6i·29-s + 8.36i·31-s − 2.44·35-s + 6.24·37-s − 4.24i·41-s − 2.02i·43-s − 2.02·47-s + 1.00·49-s + ⋯ |
L(s) = 1 | + 0.447i·5-s + 0.925i·7-s − 0.305·11-s + 0.621·13-s + 1.12i·19-s − 1.74·23-s − 0.200·25-s + 1.11i·29-s + 1.50i·31-s − 0.414·35-s + 1.02·37-s − 0.662i·41-s − 0.309i·43-s − 0.295·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.850945 + 0.932956i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.850945 + 0.932956i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
good | 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 + 1.01T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 8.36T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 8.36iT - 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 2.02iT - 43T^{2} \) |
| 47 | \( 1 + 2.02T + 47T^{2} \) |
| 53 | \( 1 - 8.48iT - 53T^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 1.43iT - 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 16.2iT - 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−10.54197612224808360574611896641, −9.911311938015333752315571841586, −8.791957865936912851337355598971, −8.189649294446659914128214192050, −7.13810095398466744309977725341, −6.06348527074510286696924795489, −5.48293923958868651527858830834, −4.08418608697107236636452735569, −3.01337219456184029619079923701, −1.79513214737889439934712424815,
0.65770743660628411228719284872, 2.28683391269252970577620295552, 3.82357996446729300782971816846, 4.52342692655964692987277173120, 5.75450749876556979284980092412, 6.62563093544108745472376876398, 7.76389022954765956526325611459, 8.271679359907809544547386422848, 9.512014642829879749954312179024, 10.04490410692576761671479511917