L(s) = 1 | + (1.99 + 1.99i)3-s − 1.09i·7-s + 4.93i·9-s + (2.33 − 2.33i)11-s + (1.80 + 1.80i)13-s + 4.93·17-s + (2.03 + 2.03i)19-s + (2.17 − 2.17i)21-s − 1.45i·23-s + (−3.84 + 3.84i)27-s + (0.707 + 0.707i)29-s − 10.1·31-s + 9.28·33-s + (4.35 − 4.35i)37-s + 7.20i·39-s + ⋯ |
L(s) = 1 | + (1.14 + 1.14i)3-s − 0.412i·7-s + 1.64i·9-s + (0.703 − 0.703i)11-s + (0.501 + 0.501i)13-s + 1.19·17-s + (0.467 + 0.467i)19-s + (0.473 − 0.473i)21-s − 0.303i·23-s + (−0.740 + 0.740i)27-s + (0.131 + 0.131i)29-s − 1.81·31-s + 1.61·33-s + (0.715 − 0.715i)37-s + 1.15i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.805530291\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.805530291\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.99 - 1.99i)T + 3iT^{2} \) |
| 7 | \( 1 + 1.09iT - 7T^{2} \) |
| 11 | \( 1 + (-2.33 + 2.33i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.80 - 1.80i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 + (-2.03 - 2.03i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.45iT - 23T^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T + 29iT^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + (-4.35 + 4.35i)T - 37iT^{2} \) |
| 41 | \( 1 - 10.2iT - 41T^{2} \) |
| 43 | \( 1 + (2.22 - 2.22i)T - 43iT^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 + (0.215 - 0.215i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.16 - 1.16i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.46 + 3.46i)T + 61iT^{2} \) |
| 67 | \( 1 + (5.04 + 5.04i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.40iT - 71T^{2} \) |
| 73 | \( 1 - 5.24iT - 73T^{2} \) |
| 79 | \( 1 - 2.61T + 79T^{2} \) |
| 83 | \( 1 + (5.67 + 5.67i)T + 83iT^{2} \) |
| 89 | \( 1 - 6.87iT - 89T^{2} \) |
| 97 | \( 1 + 3.77T + 97T^{2} \) |
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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
−9.434566661314781826524548452712, −8.931958877589818847662686606577, −8.119267432116915906103064875925, −7.43096352990879235889543986885, −6.23614325799982759250820057514, −5.30319121508488276559143385421, −4.16732975658350618458424294737, −3.66807552410595607037960282925, −2.88880479782383089479143419508, −1.40477190237938111835545520982,
1.13885118340022406265081448761, 2.06473753549833158423968378771, 3.09066133654395080120289193609, 3.85746532425475331751077584727, 5.30825551074671477580019289427, 6.16670439577228257469402114130, 7.25353162872974992377349924802, 7.50144496678463738839280755434, 8.489702385004316986260270784924, 9.075756299481641394805210266494