| L(s) = 1 | + (−1.64 + 1.51i)5-s + 2.05·7-s + 5.60·11-s − 2.63i·13-s − 2.12i·17-s − 4.87i·19-s + (4.47 − 1.71i)23-s + (0.412 − 4.98i)25-s − 1.54i·29-s − 6.05·31-s + (−3.38 + 3.11i)35-s + 0.289·37-s − 2.93i·41-s − 12.2·43-s − 0.475·47-s + ⋯ |
| L(s) = 1 | + (−0.735 + 0.677i)5-s + 0.777·7-s + 1.68·11-s − 0.729i·13-s − 0.516i·17-s − 1.11i·19-s + (0.933 − 0.358i)23-s + (0.0825 − 0.996i)25-s − 0.287i·29-s − 1.08·31-s + (−0.571 + 0.526i)35-s + 0.0475·37-s − 0.459i·41-s − 1.86·43-s − 0.0693·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.732660629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.732660629\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.64 - 1.51i)T \) |
| 23 | \( 1 + (-4.47 + 1.71i)T \) |
| good | 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 - 5.60T + 11T^{2} \) |
| 13 | \( 1 + 2.63iT - 13T^{2} \) |
| 17 | \( 1 + 2.12iT - 17T^{2} \) |
| 19 | \( 1 + 4.87iT - 19T^{2} \) |
| 29 | \( 1 + 1.54iT - 29T^{2} \) |
| 31 | \( 1 + 6.05T + 31T^{2} \) |
| 37 | \( 1 - 0.289T + 37T^{2} \) |
| 41 | \( 1 + 2.93iT - 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 0.475T + 47T^{2} \) |
| 53 | \( 1 + 11.3iT - 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 0.760iT - 61T^{2} \) |
| 67 | \( 1 - 0.897T + 67T^{2} \) |
| 71 | \( 1 - 1.57iT - 71T^{2} \) |
| 73 | \( 1 + 3.69iT - 73T^{2} \) |
| 79 | \( 1 + 13.5iT - 79T^{2} \) |
| 83 | \( 1 + 6.38iT - 83T^{2} \) |
| 89 | \( 1 - 4.36T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−8.362127762621041053467639556968, −7.39827081322903028545731036618, −6.97044444192886564738501059486, −6.27481160349098851561345457357, −5.16351705386353659364158210154, −4.51058347213541353264796104302, −3.63056233843435833818252239766, −2.94039068335995274067353880043, −1.74628302276555826972902824581, −0.53683802958069272861874600935,
1.27361492575100263493515253217, 1.71456709465951768283500173004, 3.44246288652628842410782123659, 3.96476966108297097883335384911, 4.70361634033367291093269912289, 5.45684822800529565903209517463, 6.46512946978260783415055229177, 7.07794667059133538374670259341, 7.973977675260765023979123164407, 8.480160022190894383785356400270
