L(s) = 1 | + (−0.555 − 0.831i)2-s + (−0.980 − 0.195i)3-s + (−0.382 + 0.923i)4-s + (0.831 − 0.555i)5-s + (0.382 + 0.923i)6-s + (0.980 − 0.195i)8-s + (0.923 + 0.382i)9-s + (−0.923 − 0.382i)10-s + (0.555 − 0.831i)12-s + (−0.923 + 0.382i)15-s + (−0.707 − 0.707i)16-s + (−0.195 + 0.980i)17-s + (−0.195 − 0.980i)18-s + (0.707 + 1.70i)19-s + (0.195 + 0.980i)20-s + ⋯ |
L(s) = 1 | + (−0.555 − 0.831i)2-s + (−0.980 − 0.195i)3-s + (−0.382 + 0.923i)4-s + (0.831 − 0.555i)5-s + (0.382 + 0.923i)6-s + (0.980 − 0.195i)8-s + (0.923 + 0.382i)9-s + (−0.923 − 0.382i)10-s + (0.555 − 0.831i)12-s + (−0.923 + 0.382i)15-s + (−0.707 − 0.707i)16-s + (−0.195 + 0.980i)17-s + (−0.195 − 0.980i)18-s + (0.707 + 1.70i)19-s + (0.195 + 0.980i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6669966710\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6669966710\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.555 + 0.831i)T \) |
| 3 | \( 1 + (0.980 + 0.195i)T \) |
| 5 | \( 1 + (-0.831 + 0.555i)T \) |
| 17 | \( 1 + (0.195 - 0.980i)T \) |
good | 7 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 11 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (-1.38 + 0.275i)T + (0.923 - 0.382i)T^{2} \) |
| 29 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (-0.324 + 1.63i)T + (-0.923 - 0.382i)T^{2} \) |
| 37 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 43 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + (-0.785 + 0.785i)T - iT^{2} \) |
| 53 | \( 1 + (1.81 - 0.750i)T + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.923 + 1.38i)T + (-0.382 - 0.923i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 73 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 79 | \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \) |
| 83 | \( 1 + (0.360 - 0.149i)T + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (0.382 - 0.923i)T^{2} \) |
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
−10.02538242519826405192750026797, −9.513702301090795110220739475556, −8.448456151060514160997115311067, −7.68378581287742288557841831815, −6.54336751330411878569875470658, −5.67225928656602656876738153465, −4.76730168561373197031008753282, −3.72206394089092803600865139034, −2.12957255821904708995357952197, −1.15961611182617140704787241528,
1.19101894754925677556364277854, 2.91758175097654918494215683547, 4.79370030419440285852745799904, 5.17251923733840420285089898080, 6.20140673609797837889003104114, 6.95221585229642849355105804100, 7.35013438057075793026775349158, 8.956185594880236563677767216128, 9.394510952835769548459446119723, 10.19950393601395368505164665916