L(s) = 1 | + (−0.0703 − 0.0703i)3-s + (0.562 + 2.58i)7-s − 2.99i·9-s + 0.777·11-s + (−3.93 − 3.93i)13-s + (0.982 − 0.982i)17-s + 1.14·19-s + (0.142 − 0.221i)21-s + (1.46 − 1.46i)23-s + (−0.421 + 0.421i)27-s − 4.69i·29-s − 6.45i·31-s + (−0.0546 − 0.0546i)33-s + (1.30 + 1.30i)37-s + 0.553i·39-s + ⋯ |
L(s) = 1 | + (−0.0405 − 0.0405i)3-s + (0.212 + 0.977i)7-s − 0.996i·9-s + 0.234·11-s + (−1.09 − 1.09i)13-s + (0.238 − 0.238i)17-s + 0.263·19-s + (0.0310 − 0.0482i)21-s + (0.304 − 0.304i)23-s + (−0.0810 + 0.0810i)27-s − 0.870i·29-s − 1.15i·31-s + (−0.00951 − 0.00951i)33-s + (0.214 + 0.214i)37-s + 0.0886i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.437671413\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.437671413\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.562 - 2.58i)T \) |
good | 3 | \( 1 + (0.0703 + 0.0703i)T + 3iT^{2} \) |
| 11 | \( 1 - 0.777T + 11T^{2} \) |
| 13 | \( 1 + (3.93 + 3.93i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.982 + 0.982i)T - 17iT^{2} \) |
| 19 | \( 1 - 1.14T + 19T^{2} \) |
| 23 | \( 1 + (-1.46 + 1.46i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.69iT - 29T^{2} \) |
| 31 | \( 1 + 6.45iT - 31T^{2} \) |
| 37 | \( 1 + (-1.30 - 1.30i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.81iT - 41T^{2} \) |
| 43 | \( 1 + (-7.13 + 7.13i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.34 + 7.34i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.08 + 2.08i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.29T + 59T^{2} \) |
| 61 | \( 1 + 5.88iT - 61T^{2} \) |
| 67 | \( 1 + (-6.30 - 6.30i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + (-7.23 - 7.23i)T + 73iT^{2} \) |
| 79 | \( 1 + 2.83iT - 79T^{2} \) |
| 83 | \( 1 + (10.4 + 10.4i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.41T + 89T^{2} \) |
| 97 | \( 1 + (-8.50 + 8.50i)T - 97iT^{2} \) |
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
−9.571609654194785503726209856455, −8.624560433269808060949739429715, −7.85971689227671600047107629518, −6.99173784431379110582880937113, −5.98029745300737103503890919931, −5.40220845457887361202417858020, −4.34100589512947223007893804750, −3.15007855760757494151390474492, −2.30610303347765458208087668563, −0.62816243819361895433466850009,
1.33282012519571227197449963548, 2.49886637012997781609619319405, 3.82229802692041385203912526396, 4.66047090919843146985386490878, 5.35459729009407390624755796347, 6.64877213962514989527746819691, 7.33827673006812556752057571923, 7.88763566162685557607751558300, 9.031178795583720435468660499318, 9.665274407500989472434138740633