| L(s) = 1 | − 2i·2-s − 4·4-s + 11i·7-s + 8i·8-s − 36·11-s − 17i·13-s + 22·14-s + 16·16-s − 12i·17-s + 91·19-s + 72i·22-s − 60i·23-s − 34·26-s − 44i·28-s + 276·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.593i·7-s + 0.353i·8-s − 0.986·11-s − 0.362i·13-s + 0.419·14-s + 0.250·16-s − 0.171i·17-s + 1.09·19-s + 0.697i·22-s − 0.543i·23-s − 0.256·26-s − 0.296i·28-s + 1.76·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.647289323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.647289323\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 11iT - 343T^{2} \) |
| 11 | \( 1 + 36T + 1.33e3T^{2} \) |
| 13 | \( 1 + 17iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 12iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 91T + 6.85e3T^{2} \) |
| 23 | \( 1 + 60iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 276T + 2.43e4T^{2} \) |
| 31 | \( 1 - 191T + 2.97e4T^{2} \) |
| 37 | \( 1 - 254iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 60T + 6.89e4T^{2} \) |
| 43 | \( 1 - 49iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 600iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 612iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 744T + 2.05e5T^{2} \) |
| 61 | \( 1 - 167T + 2.26e5T^{2} \) |
| 67 | \( 1 + 457iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 588T + 3.57e5T^{2} \) |
| 73 | \( 1 - 970iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 164T + 4.93e5T^{2} \) |
| 83 | \( 1 + 696iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3iT - 9.12e5T^{2} \) |
| 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
−10.29841567752806645979354562846, −9.987175152020956575362982121629, −8.664658720501297887862065302921, −8.091122402164440803502143178906, −6.78293055421717022132271180181, −5.49338182863879290846114428314, −4.74700347141400335362808966950, −3.22762415154801947909160522475, −2.38164170879179700850593233672, −0.73564221042227265882887672263,
0.929758947206530202672196434107, 2.83164762257168451306261265075, 4.19482502408834331661313655628, 5.16380232539005393820449772736, 6.18710203317889426590344945192, 7.26469945981216460894488234391, 7.88638865130561664464652003333, 8.911276888816445216283732774295, 9.934978587026581263393392126340, 10.62284928853726417795088437891
