| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 4·11-s − 2·13-s + 15-s − 2·17-s − 2·19-s − 4·21-s + 6·23-s + 25-s + 27-s + 6·29-s + 10·31-s + 4·33-s − 4·35-s − 2·39-s − 2·41-s − 2·43-s + 45-s + 4·47-s + 9·49-s − 2·51-s + 10·53-s + 4·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.458·19-s − 0.872·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.696·33-s − 0.676·35-s − 0.320·39-s − 0.312·41-s − 0.304·43-s + 0.149·45-s + 0.583·47-s + 9/7·49-s − 0.280·51-s + 1.37·53-s + 0.539·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.595851930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.595851930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.61483424616315, −12.17346567569060, −11.89553974809271, −11.28997437762865, −10.52120220395474, −10.23336488999984, −9.917815441053080, −9.220124005312769, −9.083237409882077, −8.694878845345783, −8.090859866478234, −7.364616479201117, −6.938548183077891, −6.523830333948197, −6.282751047285777, −5.715638993402365, −4.876535280364436, −4.481987004615185, −4.017666842659450, −3.230412455645528, −2.999301144246956, −2.486719697505986, −1.818383435231640, −1.052811157831913, −0.5287520451234930,
0.5287520451234930, 1.052811157831913, 1.818383435231640, 2.486719697505986, 2.999301144246956, 3.230412455645528, 4.017666842659450, 4.481987004615185, 4.876535280364436, 5.715638993402365, 6.282751047285777, 6.523830333948197, 6.938548183077891, 7.364616479201117, 8.090859866478234, 8.694878845345783, 9.083237409882077, 9.220124005312769, 9.917815441053080, 10.23336488999984, 10.52120220395474, 11.28997437762865, 11.89553974809271, 12.17346567569060, 12.61483424616315
