| L(s) = 1 | + (−2.93 + 0.447i)3-s + (1.46 − 1.73i)5-s + (−0.690 − 0.958i)7-s + (5.52 − 1.72i)9-s + (−2.12 − 2.00i)11-s + (0.538 + 4.04i)13-s + (−3.51 + 5.74i)15-s + (−0.598 − 0.900i)17-s + (−0.0363 + 0.00840i)19-s + (2.45 + 2.50i)21-s + (0.412 − 1.95i)23-s + (−0.0258 − 0.150i)25-s + (−7.43 + 3.63i)27-s + (−0.394 − 1.86i)29-s + (−6.59 − 1.26i)31-s + ⋯ |
| L(s) = 1 | + (−1.69 + 0.258i)3-s + (0.654 − 0.776i)5-s + (−0.261 − 0.362i)7-s + (1.84 − 0.575i)9-s + (−0.641 − 0.605i)11-s + (0.149 + 1.12i)13-s + (−0.906 + 1.48i)15-s + (−0.145 − 0.218i)17-s + (−0.00834 + 0.00192i)19-s + (0.535 + 0.545i)21-s + (0.0859 − 0.406i)23-s + (−0.00516 − 0.0300i)25-s + (−1.43 + 0.699i)27-s + (−0.0732 − 0.346i)29-s + (−1.18 − 0.226i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0400606 - 0.261205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0400606 - 0.261205i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + (-7.34 - 10.6i)T \) |
| good | 3 | \( 1 + (2.93 - 0.447i)T + (2.86 - 0.894i)T^{2} \) |
| 5 | \( 1 + (-1.46 + 1.73i)T + (-0.847 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.690 + 0.958i)T + (-2.21 + 6.64i)T^{2} \) |
| 11 | \( 1 + (2.12 + 2.00i)T + (0.624 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.538 - 4.04i)T + (-12.5 + 3.40i)T^{2} \) |
| 17 | \( 1 + (0.598 + 0.900i)T + (-6.57 + 15.6i)T^{2} \) |
| 19 | \( 1 + (0.0363 - 0.00840i)T + (17.0 - 8.33i)T^{2} \) |
| 23 | \( 1 + (-0.412 + 1.95i)T + (-21.0 - 9.30i)T^{2} \) |
| 29 | \( 1 + (0.394 + 1.86i)T + (-26.5 + 11.7i)T^{2} \) |
| 31 | \( 1 + (6.59 + 1.26i)T + (28.8 + 11.4i)T^{2} \) |
| 37 | \( 1 + (8.98 + 2.80i)T + (30.4 + 21.0i)T^{2} \) |
| 41 | \( 1 + (7.44 + 2.96i)T + (29.8 + 28.1i)T^{2} \) |
| 43 | \( 1 + (8.86 - 3.14i)T + (33.4 - 27.0i)T^{2} \) |
| 47 | \( 1 + (1.64 + 6.56i)T + (-41.4 + 22.2i)T^{2} \) |
| 53 | \( 1 + (7.14 - 5.36i)T + (14.8 - 50.8i)T^{2} \) |
| 59 | \( 1 + (0.837 - 1.26i)T + (-22.8 - 54.4i)T^{2} \) |
| 61 | \( 1 + (0.488 - 0.625i)T + (-14.8 - 59.1i)T^{2} \) |
| 67 | \( 1 + (-5.75 - 6.82i)T + (-11.3 + 66.0i)T^{2} \) |
| 71 | \( 1 + (-0.108 + 1.13i)T + (-69.7 - 13.3i)T^{2} \) |
| 73 | \( 1 + (1.61 + 2.88i)T + (-38.0 + 62.2i)T^{2} \) |
| 79 | \( 1 + (-14.8 + 0.563i)T + (78.7 - 5.97i)T^{2} \) |
| 83 | \( 1 + (3.21 + 0.872i)T + (71.6 + 41.9i)T^{2} \) |
| 89 | \( 1 + (-0.957 - 1.56i)T + (-40.5 + 79.2i)T^{2} \) |
| 97 | \( 1 + (9.53 - 1.82i)T + (90.1 - 35.8i)T^{2} \) |
| 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
−10.27902004331676236763065685964, −9.468352505260647306841698142138, −8.577335436316050038552167589055, −7.10139189764773269569376983134, −6.40246037076389107688451745353, −5.42684634702339569694150715521, −4.96142742457114680231817272394, −3.78753805048742603883834136428, −1.67935762841431840915204376833, −0.17182133735909938775218001555,
1.76782393015884729208316505240, 3.24139775578602491509707368890, 4.99185100605990150221063520985, 5.50899708790433351294669232398, 6.42232158868506001868662377807, 7.00029242038049000077913546077, 8.089961787706311370636416366134, 9.565226182972242469172984651978, 10.41643512038563717926258570229, 10.70769858381800249560761675268
