L(s) = 1 | + (−0.981 − 1.42i)3-s + (0.276 + 0.479i)5-s + (−1.07 + 2.80i)9-s + (−4.03 − 2.32i)11-s + (3.58 + 2.06i)13-s + (0.412 − 0.866i)15-s + (3.62 + 6.27i)17-s + (−5.81 − 3.35i)19-s + (4.85 − 2.80i)23-s + (2.34 − 4.06i)25-s + (5.05 − 1.22i)27-s + (1.16 − 0.673i)29-s + 0.959i·31-s + (0.637 + 8.04i)33-s + (3.53 − 6.12i)37-s + ⋯ |
L(s) = 1 | + (−0.566 − 0.823i)3-s + (0.123 + 0.214i)5-s + (−0.357 + 0.933i)9-s + (−1.21 − 0.702i)11-s + (0.993 + 0.573i)13-s + (0.106 − 0.223i)15-s + (0.878 + 1.52i)17-s + (−1.33 − 0.770i)19-s + (1.01 − 0.584i)23-s + (0.469 − 0.812i)25-s + (0.972 − 0.234i)27-s + (0.216 − 0.125i)29-s + 0.172i·31-s + (0.110 + 1.40i)33-s + (0.581 − 1.00i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.262141641\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.262141641\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.981 + 1.42i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.276 - 0.479i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (4.03 + 2.32i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.58 - 2.06i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.62 - 6.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.81 + 3.35i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.85 + 2.80i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.16 + 0.673i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 0.959iT - 31T^{2} \) |
| 37 | \( 1 + (-3.53 + 6.12i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.39 - 4.14i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.02 + 1.78i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.80T + 47T^{2} \) |
| 53 | \( 1 + (-7.30 + 4.21i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 7.79T + 59T^{2} \) |
| 61 | \( 1 + 6.20iT - 61T^{2} \) |
| 67 | \( 1 + 3.37T + 67T^{2} \) |
| 71 | \( 1 - 0.407iT - 71T^{2} \) |
| 73 | \( 1 + (-7.47 + 4.31i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 0.636T + 79T^{2} \) |
| 83 | \( 1 + (-2.78 - 4.82i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.46 + 6.00i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.48 + 4.32i)T + (48.5 - 84.0i)T^{2} \) |
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
−8.770899765743630730098728366496, −8.402593889072541450224130175088, −7.57007178872674422441871423934, −6.53206382062545997706278121080, −6.13669849752033948931410119840, −5.27353285547203460125903083813, −4.23059627044620679347955322316, −2.95436557990060835519312106465, −1.98844458095156198311768873041, −0.65274609129038658827933857311,
0.984312648078076540740710403171, 2.69492841778928397331597237469, 3.59164318317573082554713250471, 4.70737093982098740493011149440, 5.30224548394172165114892940586, 5.97382401816736220123412783428, 7.06809325202093190896616083179, 7.88577382079863155578652925104, 8.840012300303324967028482217759, 9.488380686812677351912134136190