L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (−1.12 − 0.541i)5-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (0.777 + 0.974i)10-s + (0.400 − 1.75i)13-s + (−0.222 + 0.974i)16-s − 0.445·17-s + (−0.222 + 0.974i)18-s + (−0.277 − 1.21i)20-s + (0.346 + 0.433i)25-s + (−1.12 + 1.40i)26-s + (0.623 − 0.781i)32-s + (0.400 + 0.193i)34-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (−1.12 − 0.541i)5-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (0.777 + 0.974i)10-s + (0.400 − 1.75i)13-s + (−0.222 + 0.974i)16-s − 0.445·17-s + (−0.222 + 0.974i)18-s + (−0.277 − 1.21i)20-s + (0.346 + 0.433i)25-s + (−1.12 + 1.40i)26-s + (0.623 − 0.781i)32-s + (0.400 + 0.193i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3077338230\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3077338230\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 5 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 7 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 11 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 23 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 37 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + 1.80T + T^{2} \) |
| 43 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 67 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \) |
| 79 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 97 | \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
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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
−8.227441026601335291435474999510, −8.150779302424272969165041429214, −7.08928494035303301756416808337, −6.39484754227545922999969332313, −5.36793043895408701707158221870, −4.25600065812498450404180649119, −3.45552595057050463494415096464, −2.88674984978711218295384021648, −1.29477823899248074335395728317, −0.26308499647106882150646856266,
1.67688352348328040367087351093, 2.56301078166903564633845327095, 3.80355958531023798282039366291, 4.60857922638759206007218371242, 5.57529044940820766242727856851, 6.58697847916443316207385135705, 7.05454206531847552398918968225, 7.74634717738239655760282443625, 8.380995483927912150344196917529, 9.029229395925608544990360899619