L(s) = 1 | − 2.14·3-s − 2.61i·5-s + 1.58·9-s + 3.95i·11-s − 1.08i·13-s + 5.59i·15-s + 0.317i·17-s − 5.16·19-s − 2.31i·23-s − 1.82·25-s + 3.02·27-s + 6.82·29-s + 6.05·31-s − 8.47i·33-s + 4·37-s + ⋯ |
L(s) = 1 | − 1.23·3-s − 1.16i·5-s + 0.528·9-s + 1.19i·11-s − 0.300i·13-s + 1.44i·15-s + 0.0768i·17-s − 1.18·19-s − 0.483i·23-s − 0.365·25-s + 0.582·27-s + 1.26·29-s + 1.08·31-s − 1.47i·33-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7975467068\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7975467068\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2.14T + 3T^{2} \) |
| 5 | \( 1 + 2.61iT - 5T^{2} \) |
| 11 | \( 1 - 3.95iT - 11T^{2} \) |
| 13 | \( 1 + 1.08iT - 13T^{2} \) |
| 17 | \( 1 - 0.317iT - 17T^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 23 | \( 1 + 2.31iT - 23T^{2} \) |
| 29 | \( 1 - 6.82T + 29T^{2} \) |
| 31 | \( 1 - 6.05T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 2.29iT - 41T^{2} \) |
| 43 | \( 1 - 7.23iT - 43T^{2} \) |
| 47 | \( 1 + 4.28T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 5.41iT - 61T^{2} \) |
| 67 | \( 1 + 3.27iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 14.0iT - 73T^{2} \) |
| 79 | \( 1 - 7.91iT - 79T^{2} \) |
| 83 | \( 1 - 9.45T + 83T^{2} \) |
| 89 | \( 1 - 5.99iT - 89T^{2} \) |
| 97 | \( 1 - 9.23iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.399441594770563439196905996326, −7.87219638927341898079714179119, −6.56435809170821918012713286746, −6.37662379514331281768635068375, −5.20852382606521577160080026680, −4.76499002147272092024345064171, −4.22193232445292282119348081359, −2.65492298071830704570442344794, −1.43439989388875242403955704948, −0.39566879361475298954293762748,
0.901712979031246792847353060272, 2.46150210732057981425940235589, 3.27331356086466243900511380116, 4.31591534536871193478291185631, 5.20392482791887495075799017162, 6.14414697696848352704688863670, 6.39920358163851396473427490643, 7.08821254064153995476984084848, 8.171437521172317144693689467111, 8.762304448504709644810281850125