| L(s) = 1 | + (−0.557 + 4.96i)5-s − 10.7i·7-s − 12.1i·11-s + 23.3i·13-s + 1.67·17-s + 0.234·19-s + 19.8·23-s + (−24.3 − 5.53i)25-s − 34.0i·29-s − 19.0·31-s + (53.2 + 5.97i)35-s − 32.7i·37-s − 5.27i·41-s + 52.9i·43-s + 53.7·47-s + ⋯ |
| L(s) = 1 | + (−0.111 + 0.993i)5-s − 1.53i·7-s − 1.10i·11-s + 1.79i·13-s + 0.0987·17-s + 0.0123·19-s + 0.863·23-s + (−0.975 − 0.221i)25-s − 1.17i·29-s − 0.615·31-s + (1.52 + 0.170i)35-s − 0.884i·37-s − 0.128i·41-s + 1.23i·43-s + 1.14·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.247635151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.247635151\) |
| \(L(2)\) |
|
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 | \( 1 + (0.557 - 4.96i)T \) |
| good | 7 | \( 1 + 10.7iT - 49T^{2} \) |
| 11 | \( 1 + 12.1iT - 121T^{2} \) |
| 13 | \( 1 - 23.3iT - 169T^{2} \) |
| 17 | \( 1 - 1.67T + 289T^{2} \) |
| 19 | \( 1 - 0.234T + 361T^{2} \) |
| 23 | \( 1 - 19.8T + 529T^{2} \) |
| 29 | \( 1 + 34.0iT - 841T^{2} \) |
| 31 | \( 1 + 19.0T + 961T^{2} \) |
| 37 | \( 1 + 32.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 5.27iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 52.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 53.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 84.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 88.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 65.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 99.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 36.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 79.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 35.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 27.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 152. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 97.0iT - 9.40e3T^{2} \) |
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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
−9.149927128206563375637055832112, −7.988975978367313727576737870957, −7.32847250766503987438279700449, −6.65632385464186485676250298760, −6.01848120843375779018334622388, −4.57527398479245874805938032337, −3.87934223222175175733237854583, −3.08208454662877084623167358760, −1.73550654821694212247722113260, −0.35621485886260789948300474528,
1.19615595567139027450460214466, 2.37998702593284134502455671243, 3.33885861317239593815389109229, 4.71253726444240325160063728491, 5.31873715377972392091990043406, 5.87054296084911601930967159589, 7.16052237954615288779900470370, 7.964574896764828305945565146399, 8.740740902535343629327486828157, 9.214026504046611866025325823347
