| L(s) = 1 | + (−1.56 + 2.11i)2-s + (−2.83 + 0.536i)3-s + (−1.45 − 4.72i)4-s + (1.75 + 1.38i)5-s + (3.29 − 6.85i)6-s + (2.30 − 1.30i)7-s + (7.31 + 2.55i)8-s + (4.96 − 1.94i)9-s + (−5.68 + 1.53i)10-s + (1.23 − 3.14i)11-s + (6.66 + 12.6i)12-s + (−2.93 − 0.330i)13-s + (−0.843 + 6.91i)14-s + (−5.71 − 3.00i)15-s + (−8.69 + 5.92i)16-s + (4.54 + 0.170i)17-s + ⋯ |
| L(s) = 1 | + (−1.10 + 1.49i)2-s + (−1.63 + 0.309i)3-s + (−0.728 − 2.36i)4-s + (0.783 + 0.621i)5-s + (1.34 − 2.79i)6-s + (0.870 − 0.492i)7-s + (2.58 + 0.904i)8-s + (1.65 − 0.649i)9-s + (−1.79 + 0.486i)10-s + (0.372 − 0.949i)11-s + (1.92 + 3.63i)12-s + (−0.812 − 0.0915i)13-s + (−0.225 + 1.84i)14-s + (−1.47 − 0.775i)15-s + (−2.17 + 1.48i)16-s + (1.10 + 0.0412i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0825 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0825 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.344023 + 0.373688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.344023 + 0.373688i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.75 - 1.38i)T \) |
| 7 | \( 1 + (-2.30 + 1.30i)T \) |
| good | 2 | \( 1 + (1.56 - 2.11i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (2.83 - 0.536i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-1.23 + 3.14i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (2.93 + 0.330i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (-4.54 - 0.170i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (0.821 - 1.42i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0691 + 1.84i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-4.06 + 0.927i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-3.56 + 2.05i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.02 - 0.543i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-5.06 - 10.5i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (1.33 + 3.82i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (0.836 + 0.617i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-3.65 - 1.93i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.814 - 10.8i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (0.253 - 0.823i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-3.02 - 0.810i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.98 + 13.0i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-3.57 + 2.63i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-0.541 - 0.312i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.315 + 2.80i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-2.31 - 5.90i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (8.62 + 8.62i)T + 97iT^{2} \) |
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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
−11.96413758421961391084761908921, −10.90790453061295268435421440608, −10.30905944425941929228966889332, −9.634955414483654166192533120684, −8.213646917931333535888617591379, −7.18447088480622346039893527853, −6.26809682033332971726316279394, −5.65761397066032273649679873699, −4.72333800267582417351202724811, −1.01376672367512473679909028814,
1.08406318965012643319927711729, 2.15703974164368205573061802279, 4.54350125523790327593681904396, 5.40781161225395216351778595749, 7.00852245443892492944320950942, 8.187691703433605789267787615051, 9.408886096476155782412814917223, 10.07720250101674036102379305126, 10.91468164767240133255979133049, 11.93351573088878974985097872286
