L(s) = 1 | + (0.433 + 0.750i)5-s + (2.32 + 1.26i)7-s + (1.75 − 3.04i)11-s + (0.933 − 1.61i)13-s + (3.25 + 5.64i)17-s + (2.69 − 4.66i)19-s + (−4.32 − 7.48i)23-s + (2.12 − 3.67i)25-s + (−1.75 − 3.04i)29-s − 1.86·31-s + (0.0576 + 2.29i)35-s + (1.39 − 2.40i)37-s + (−5.19 + 8.99i)41-s + (2.89 + 5.00i)43-s + 6.16·47-s + ⋯ |
L(s) = 1 | + (0.193 + 0.335i)5-s + (0.878 + 0.478i)7-s + (0.529 − 0.917i)11-s + (0.258 − 0.448i)13-s + (0.790 + 1.36i)17-s + (0.617 − 1.06i)19-s + (−0.901 − 1.56i)23-s + (0.424 − 0.735i)25-s + (−0.326 − 0.565i)29-s − 0.335·31-s + (0.00975 + 0.387i)35-s + (0.228 − 0.395i)37-s + (−0.810 + 1.40i)41-s + (0.440 + 0.763i)43-s + 0.898·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.221268796\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.221268796\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.32 - 1.26i)T \) |
good | 5 | \( 1 + (-0.433 - 0.750i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.75 + 3.04i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.933 + 1.61i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.25 - 5.64i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.69 + 4.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.32 + 7.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.75 + 3.04i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 + (-1.39 + 2.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.19 - 8.99i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.89 - 5.00i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6.16T + 47T^{2} \) |
| 53 | \( 1 + (2.80 + 4.85i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 5.64T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 1.35T + 67T^{2} \) |
| 71 | \( 1 - 2.08T + 71T^{2} \) |
| 73 | \( 1 + (3.62 + 6.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + (3.43 + 5.94i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.28 + 5.68i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.64 - 2.85i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.781729242647865469709510851492, −8.292265190169356939491917230487, −7.64747233684060161532905329399, −6.36424456138633843075677036952, −6.04229050583594634108255234169, −5.05093489572220188866976922867, −4.12994541311421590259753821535, −3.12584392163897679248954855784, −2.15715386940423920063301910394, −0.914044082635881364894824069893,
1.21606995324140912226646053161, 1.91231108368432095931619442053, 3.45597806276395045500000423571, 4.17533447331124647999269684518, 5.22607629042674384090148859654, 5.61561621194536274364643144542, 7.08575727976752613936693332841, 7.35938879593374639905666131060, 8.208661871914611398297090651962, 9.252319541354198811409339615272