| L(s) = 1 | − 1.73i·7-s − 3.46i·11-s − 4.58i·13-s − 3·17-s + (−2.64 + 3.46i)19-s − 5.19i·23-s − 5·25-s + 4.58i·29-s − 5.29·31-s + 9.16i·37-s + 9.16i·41-s + 3.46i·47-s + 4·49-s − 4.58i·53-s − 7.93·59-s + ⋯ |
| L(s) = 1 | − 0.654i·7-s − 1.04i·11-s − 1.27i·13-s − 0.727·17-s + (−0.606 + 0.794i)19-s − 1.08i·23-s − 25-s + 0.850i·29-s − 0.950·31-s + 1.50i·37-s + 1.43i·41-s + 0.505i·47-s + 0.571·49-s − 0.629i·53-s − 1.03·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3932360554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3932360554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (2.64 - 3.46i)T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 1.73iT - 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 4.58iT - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 + 5.19iT - 23T^{2} \) |
| 29 | \( 1 - 4.58iT - 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 - 9.16iT - 37T^{2} \) |
| 41 | \( 1 - 9.16iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 4.58iT - 53T^{2} \) |
| 59 | \( 1 + 7.93T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 + 6.92iT - 83T^{2} \) |
| 89 | \( 1 - 18.3iT - 89T^{2} \) |
| 97 | \( 1 + 9.16iT - 97T^{2} \) |
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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
−8.151337421284939889397191520681, −8.003324726015243498065899352499, −6.80551505790772345544861650278, −6.17541702756754149040449313787, −5.37689798950933125439467544931, −4.41556998186001713596199870720, −3.55785496832310287662216128186, −2.75032827747300981486839392581, −1.38183004231919183301121952735, −0.12157138042056293530031649410,
1.90502802454233838702577280717, 2.30924569151904736462449015136, 3.82872867498499349140443222270, 4.41263932494147596938089858846, 5.38082830552768655783380532580, 6.15337043931015113580391178321, 7.06462084535370256240337159569, 7.51536451639472427752502780497, 8.662202686253431384057628136519, 9.269355637984452555818981947820
