| L(s) = 1 | + (1.26 − 4.83i)5-s − 6.26i·7-s − 1.03i·11-s + 3i·13-s + 15.7·17-s − 18.7·19-s − 34.6·23-s + (−21.7 − 12.2i)25-s + 3.10i·29-s + 17.7·31-s + (−30.2 − 7.93i)35-s − 43.3i·37-s − 46.2i·41-s − 20.7i·43-s − 78.8·47-s + ⋯ |
| L(s) = 1 | + (0.253 − 0.967i)5-s − 0.894i·7-s − 0.0939i·11-s + 0.230i·13-s + 0.928·17-s − 0.988·19-s − 1.50·23-s + (−0.871 − 0.490i)25-s + 0.106i·29-s + 0.573·31-s + (−0.865 − 0.226i)35-s − 1.17i·37-s − 1.12i·41-s − 0.482i·43-s − 1.67·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7650966971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7650966971\) |
| \(L(2)\) |
|
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 | \( 1 + (-1.26 + 4.83i)T \) |
| good | 7 | \( 1 + 6.26iT - 49T^{2} \) |
| 11 | \( 1 + 1.03iT - 121T^{2} \) |
| 13 | \( 1 - 3iT - 169T^{2} \) |
| 17 | \( 1 - 15.7T + 289T^{2} \) |
| 19 | \( 1 + 18.7T + 361T^{2} \) |
| 23 | \( 1 + 34.6T + 529T^{2} \) |
| 29 | \( 1 - 3.10iT - 841T^{2} \) |
| 31 | \( 1 - 17.7T + 961T^{2} \) |
| 37 | \( 1 + 43.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 46.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 20.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 78.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 44.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 90.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 8.78T + 3.72e3T^{2} \) |
| 67 | \( 1 + 19.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 56.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 109. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 39T + 6.24e3T^{2} \) |
| 83 | \( 1 + 75.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 90.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 8.88iT - 9.40e3T^{2} \) |
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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
−8.440140476061968082775339439284, −7.82152500207037695474539411098, −6.96050316421994135536529478928, −6.00525778400735154194665266532, −5.32151099602042691305631362699, −4.24967204157105624026619759918, −3.81686539693416279218864018701, −2.30264052003163755350470454166, −1.27165227803762381339487405192, −0.18449798150620250233370975534,
1.67107523065236679471671337533, 2.60130978808000586407007034539, 3.37178317240982162256946967457, 4.47390291755832488259864157595, 5.55608255360036553337657265925, 6.19421943257475548863900411589, 6.80370919225820020309093751068, 7.992702728809391904888866140176, 8.286897030580318719143947230591, 9.526217657091120337764332486270
