L(s) = 1 | + (−2.96 + 2.96i)5-s − 7-s + (−0.569 − 0.569i)11-s + (2.53 − 2.53i)13-s − 1.03i·17-s + (5.23 + 5.23i)19-s + 8.71i·23-s − 12.6i·25-s + (6.05 + 6.05i)29-s + 3.00i·31-s + (2.96 − 2.96i)35-s + (−0.149 − 0.149i)37-s − 8.63·41-s + (−1.73 + 1.73i)43-s + 4.10·47-s + ⋯ |
L(s) = 1 | + (−1.32 + 1.32i)5-s − 0.377·7-s + (−0.171 − 0.171i)11-s + (0.703 − 0.703i)13-s − 0.251i·17-s + (1.20 + 1.20i)19-s + 1.81i·23-s − 2.52i·25-s + (1.12 + 1.12i)29-s + 0.539i·31-s + (0.501 − 0.501i)35-s + (−0.0246 − 0.0246i)37-s − 1.34·41-s + (−0.264 + 0.264i)43-s + 0.599·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8135024106\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8135024106\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (2.96 - 2.96i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.569 + 0.569i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.53 + 2.53i)T - 13iT^{2} \) |
| 17 | \( 1 + 1.03iT - 17T^{2} \) |
| 19 | \( 1 + (-5.23 - 5.23i)T + 19iT^{2} \) |
| 23 | \( 1 - 8.71iT - 23T^{2} \) |
| 29 | \( 1 + (-6.05 - 6.05i)T + 29iT^{2} \) |
| 31 | \( 1 - 3.00iT - 31T^{2} \) |
| 37 | \( 1 + (0.149 + 0.149i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.63T + 41T^{2} \) |
| 43 | \( 1 + (1.73 - 1.73i)T - 43iT^{2} \) |
| 47 | \( 1 - 4.10T + 47T^{2} \) |
| 53 | \( 1 + (-6.04 + 6.04i)T - 53iT^{2} \) |
| 59 | \( 1 + (6.05 + 6.05i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5.81 + 5.81i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.0256 + 0.0256i)T + 67iT^{2} \) |
| 71 | \( 1 - 14.5iT - 71T^{2} \) |
| 73 | \( 1 - 5.38iT - 73T^{2} \) |
| 79 | \( 1 - 3.89iT - 79T^{2} \) |
| 83 | \( 1 + (1.40 - 1.40i)T - 83iT^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 3.75T + 97T^{2} \) |
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
−8.399454697957408693050388929171, −8.093793857346096537966372420579, −7.17638654372050024683528039604, −6.88172042469598093019765492389, −5.84335764241466487549793464199, −5.13855820668511439902613752299, −3.76871640819167478986065485329, −3.44802445301509332759414925009, −2.84042493970552985669678212154, −1.23610662840399962058062311943,
0.28940014217040018026093584116, 1.14979752466374324111892401893, 2.60456284713814329114846353344, 3.63604090032493666242230341802, 4.44041665814572809233215099539, 4.78932784648779857421847950654, 5.84998619846809768336748075359, 6.76741977923826333402850150062, 7.44888312569580585645503218335, 8.269837395721747882645131179221