L(s) = 1 | + (−0.707 + 0.707i)3-s + (2.49 − 0.884i)7-s − 1.00i·9-s + 3.27·11-s + (−3.02 + 3.02i)13-s + (1.41 + 1.41i)17-s + 2.15·19-s + (−1.13 + 2.38i)21-s + (−0.296 − 0.296i)23-s + (0.707 + 0.707i)27-s + 0.725i·29-s + 1.31i·31-s + (−2.31 + 2.31i)33-s + (1.56 − 1.56i)37-s − 4.27i·39-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.942 − 0.334i)7-s − 0.333i·9-s + 0.987·11-s + (−0.838 + 0.838i)13-s + (0.342 + 0.342i)17-s + 0.493·19-s + (−0.248 + 0.521i)21-s + (−0.0617 − 0.0617i)23-s + (0.136 + 0.136i)27-s + 0.134i·29-s + 0.235i·31-s + (−0.403 + 0.403i)33-s + (0.256 − 0.256i)37-s − 0.684i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.769788873\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.769788873\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.49 + 0.884i)T \) |
good | 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 + (3.02 - 3.02i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.15T + 19T^{2} \) |
| 23 | \( 1 + (0.296 + 0.296i)T + 23iT^{2} \) |
| 29 | \( 1 - 0.725iT - 29T^{2} \) |
| 31 | \( 1 - 1.31iT - 31T^{2} \) |
| 37 | \( 1 + (-1.56 + 1.56i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.25iT - 41T^{2} \) |
| 43 | \( 1 + (-0.632 - 0.632i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.71 - 3.71i)T + 53iT^{2} \) |
| 59 | \( 1 - 8.71T + 59T^{2} \) |
| 61 | \( 1 + 1.31iT - 61T^{2} \) |
| 67 | \( 1 + (3.08 - 3.08i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.27T + 71T^{2} \) |
| 73 | \( 1 + (7.07 - 7.07i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.8iT - 79T^{2} \) |
| 83 | \( 1 + (11.7 - 11.7i)T - 83iT^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 + (-7.26 - 7.26i)T + 97iT^{2} \) |
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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
−9.212881443856790588509256516084, −8.535990424122219315315277891014, −7.49940252412015295912071382242, −6.93557705034190714672742501884, −5.95544245368008238528700848798, −5.08720579896370528838928850415, −4.35399444309918797199265722524, −3.64750471588083753451964860399, −2.18619364500423764764685319391, −1.06778874172848336327001013043,
0.837788799099623172179981282775, 1.94906990932597057348923078201, 3.04045268117022835787778845989, 4.28171118249497349742630711097, 5.14050210556050786737308956522, 5.74878393533083438164091590899, 6.72216737101234896567150389466, 7.53030592324612135496571990119, 8.089870092322129181585195822161, 8.985482552279118974100863289827