L(s) = 1 | + 1.96·3-s + (−1.60 − 1.60i)7-s + 0.851·9-s + (−0.754 + 0.754i)11-s − 5.94i·13-s + (−1.95 − 1.95i)17-s + (−0.780 + 0.780i)19-s + (−3.14 − 3.14i)21-s + (4.93 − 4.93i)23-s − 4.21·27-s + (−1.44 − 1.44i)29-s + 3.60i·31-s + (−1.48 + 1.48i)33-s − 10.2i·37-s − 11.6i·39-s + ⋯ |
L(s) = 1 | + 1.13·3-s + (−0.605 − 0.605i)7-s + 0.283·9-s + (−0.227 + 0.227i)11-s − 1.64i·13-s + (−0.474 − 0.474i)17-s + (−0.179 + 0.179i)19-s + (−0.686 − 0.686i)21-s + (1.02 − 1.02i)23-s − 0.811·27-s + (−0.268 − 0.268i)29-s + 0.648i·31-s + (−0.257 + 0.257i)33-s − 1.68i·37-s − 1.86i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.783817543\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.783817543\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 1.96T + 3T^{2} \) |
| 7 | \( 1 + (1.60 + 1.60i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.754 - 0.754i)T - 11iT^{2} \) |
| 13 | \( 1 + 5.94iT - 13T^{2} \) |
| 17 | \( 1 + (1.95 + 1.95i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.780 - 0.780i)T - 19iT^{2} \) |
| 23 | \( 1 + (-4.93 + 4.93i)T - 23iT^{2} \) |
| 29 | \( 1 + (1.44 + 1.44i)T + 29iT^{2} \) |
| 31 | \( 1 - 3.60iT - 31T^{2} \) |
| 37 | \( 1 + 10.2iT - 37T^{2} \) |
| 41 | \( 1 + 6.93iT - 41T^{2} \) |
| 43 | \( 1 - 9.91iT - 43T^{2} \) |
| 47 | \( 1 + (-0.104 + 0.104i)T - 47iT^{2} \) |
| 53 | \( 1 - 4.03T + 53T^{2} \) |
| 59 | \( 1 + (-3.46 - 3.46i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.680 + 0.680i)T - 61iT^{2} \) |
| 67 | \( 1 + 9.04iT - 67T^{2} \) |
| 71 | \( 1 - 3.64T + 71T^{2} \) |
| 73 | \( 1 + (-2.94 - 2.94i)T + 73iT^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 4.23T + 83T^{2} \) |
| 89 | \( 1 - 0.0426T + 89T^{2} \) |
| 97 | \( 1 + (-1.91 - 1.91i)T + 97iT^{2} \) |
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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
−9.061176576999422791960149588456, −8.476814126668846806501090626333, −7.62839673559169062787424785552, −7.06371278568676327608583024730, −5.96156491310928835753816144254, −4.98473628339140962442292691934, −3.84774596389166772466575890280, −3.07526775708202024220071700012, −2.33534326247275415005372606428, −0.56426023197238445531546334508,
1.74684382625089251199861283553, 2.67424586162040054492325456581, 3.49613312943973142475407136095, 4.43382459793018376826980924520, 5.58849381096171278296311114133, 6.53205940224633402781306588401, 7.24306295253365215031827630919, 8.279520293370815066662035053120, 8.906661460212604262178286933010, 9.333273245534729043622044398964