L(s) = 1 | + i·5-s + 1.11i·7-s − 5.37i·11-s + (−3.37 + 1.26i)13-s + 7.90i·19-s + 6.49·23-s − 25-s − 3.63·29-s − 3.14i·31-s − 1.11·35-s + 7.40i·37-s + 4.75i·41-s − 4.78·43-s + 6.16i·47-s + 5.75·49-s + ⋯ |
L(s) = 1 | + 0.447i·5-s + 0.421i·7-s − 1.62i·11-s + (−0.936 + 0.349i)13-s + 1.81i·19-s + 1.35·23-s − 0.200·25-s − 0.675·29-s − 0.565i·31-s − 0.188·35-s + 1.21i·37-s + 0.742i·41-s − 0.729·43-s + 0.898i·47-s + 0.822·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.130523553\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.130523553\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (3.37 - 1.26i)T \) |
good | 7 | \( 1 - 1.11iT - 7T^{2} \) |
| 11 | \( 1 + 5.37iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7.90iT - 19T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 + 3.63T + 29T^{2} \) |
| 31 | \( 1 + 3.14iT - 31T^{2} \) |
| 37 | \( 1 - 7.40iT - 37T^{2} \) |
| 41 | \( 1 - 4.75iT - 41T^{2} \) |
| 43 | \( 1 + 4.78T + 43T^{2} \) |
| 47 | \( 1 - 6.16iT - 47T^{2} \) |
| 53 | \( 1 + 0.292T + 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 - 11.8iT - 67T^{2} \) |
| 71 | \( 1 + 3.43iT - 71T^{2} \) |
| 73 | \( 1 - 4.59iT - 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 7.40iT - 83T^{2} \) |
| 89 | \( 1 + 14.8iT - 89T^{2} \) |
| 97 | \( 1 + 3.76iT - 97T^{2} \) |
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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
−9.177933470235311047370541385959, −8.442022078840484266156206719779, −7.75364555770641062303772813716, −6.89758844497474484123447266131, −5.98577627233457583211706540232, −5.51944016582783977251173370588, −4.37462638138323592390887030435, −3.34051081124038372810486673851, −2.67650521428190262149444669977, −1.33110058829819225830442540080,
0.39019458919118241315971819499, 1.84506365372706266855900247477, 2.79271107119581150177020004287, 4.02234230883297574561943222161, 4.94209605509107028142697276904, 5.20465622858239624411888189237, 6.76161743533042288445542275115, 7.15449111407369263542631075414, 7.80854874234542181812431189981, 9.051535267141388455035526675499