L(s) = 1 | + (1 + 1.73i)3-s + (0.5 + 0.866i)5-s + 4·7-s + (−0.499 + 0.866i)9-s + 3·11-s + (−3 + 5.19i)13-s + (−0.999 + 1.73i)15-s + (−1 − 1.73i)17-s + (−3.5 + 2.59i)19-s + (4 + 6.92i)21-s + (2 − 3.46i)23-s + (−0.499 + 0.866i)25-s + 4.00·27-s + (−0.5 + 0.866i)29-s + 5·31-s + ⋯ |
L(s) = 1 | + (0.577 + 0.999i)3-s + (0.223 + 0.387i)5-s + 1.51·7-s + (−0.166 + 0.288i)9-s + 0.904·11-s + (−0.832 + 1.44i)13-s + (−0.258 + 0.447i)15-s + (−0.242 − 0.420i)17-s + (−0.802 + 0.596i)19-s + (0.872 + 1.51i)21-s + (0.417 − 0.722i)23-s + (−0.0999 + 0.173i)25-s + 0.769·27-s + (−0.0928 + 0.160i)29-s + 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.606915778\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.606915778\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (3.5 - 2.59i)T \) |
good | 3 | \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.5 - 12.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6 + 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + (8.5 - 14.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6 + 10.3i)T + (-48.5 + 84.0i)T^{2} \) |
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
−9.665108650282069722653549891607, −8.778270963584984394363042247224, −8.448103209241532077401077917019, −7.16357483589614806200098256720, −6.58914289596137312124726065867, −5.23147247484025996508583531477, −4.41385406118851039570089918388, −3.98875876297654614203503397045, −2.58281633659935237285442425199, −1.63856205374735701460078038991,
1.07006181674947654735909188159, 1.90481701583711419969009404120, 2.87151794214658261779283704543, 4.35740711008743360675183931725, 5.06529070346488421490569925502, 6.05698972656116033746548650923, 7.14004226091595057598861284147, 7.72460010183171537284386502854, 8.411201090297281374490953571761, 8.921335094335021783608039767149