L(s) = 1 | + (0.707 − 0.707i)3-s + (−0.770 − 2.09i)5-s − 3.05·7-s − 1.00i·9-s + (1.80 − 1.80i)11-s + (2.47 − 2.47i)13-s + (−2.02 − 0.939i)15-s − 3.66i·17-s + (2.31 + 2.31i)19-s + (−2.15 + 2.15i)21-s − 4.86·23-s + (−3.81 + 3.23i)25-s + (−0.707 − 0.707i)27-s + (−4.74 − 4.74i)29-s + 1.86·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.344 − 0.938i)5-s − 1.15·7-s − 0.333i·9-s + (0.545 − 0.545i)11-s + (0.685 − 0.685i)13-s + (−0.523 − 0.242i)15-s − 0.889i·17-s + (0.531 + 0.531i)19-s + (−0.470 + 0.470i)21-s − 1.01·23-s + (−0.762 + 0.646i)25-s + (−0.136 − 0.136i)27-s + (−0.881 − 0.881i)29-s + 0.334·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9694069533\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9694069533\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.770 + 2.09i)T \) |
good | 7 | \( 1 + 3.05T + 7T^{2} \) |
| 11 | \( 1 + (-1.80 + 1.80i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.47 + 2.47i)T - 13iT^{2} \) |
| 17 | \( 1 + 3.66iT - 17T^{2} \) |
| 19 | \( 1 + (-2.31 - 2.31i)T + 19iT^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 + (4.74 + 4.74i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.86T + 31T^{2} \) |
| 37 | \( 1 + (-5.40 - 5.40i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.47iT - 41T^{2} \) |
| 43 | \( 1 + (4.19 + 4.19i)T + 43iT^{2} \) |
| 47 | \( 1 - 8.24iT - 47T^{2} \) |
| 53 | \( 1 + (9.99 + 9.99i)T + 53iT^{2} \) |
| 59 | \( 1 + (-2.47 + 2.47i)T - 59iT^{2} \) |
| 61 | \( 1 + (8.01 + 8.01i)T + 61iT^{2} \) |
| 67 | \( 1 + (8.60 - 8.60i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.63iT - 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + (-3.65 + 3.65i)T - 83iT^{2} \) |
| 89 | \( 1 + 13.1iT - 89T^{2} \) |
| 97 | \( 1 + 12.4iT - 97T^{2} \) |
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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
−8.730647385501251329017621632130, −8.096785009696804291248579420183, −7.43024257724629568989528799442, −6.25418668891811583312688261208, −5.88071116804134284675008966034, −4.63072559838534661108431577649, −3.61590662460090477323498300632, −3.01756933912406204115707584904, −1.47448350845831485621940893115, −0.33324034229914240518379382247,
1.83820902348387544015305280001, 3.01043115624110920808529676305, 3.72731203951521803190116451711, 4.33947563177771443219136345937, 5.82965976735252413875351408128, 6.47603972062724875782476477238, 7.17714722790758480574596353468, 7.970219109033506659279274417929, 9.074347346713279678364500401476, 9.457029902343393710088343628646