L(s) = 1 | + (0.745 − 1.29i)3-s + (−0.5 + 0.866i)5-s + 2.84·7-s + (0.387 + 0.670i)9-s + 0.864·11-s + (−0.321 − 0.557i)13-s + (0.745 + 1.29i)15-s + (−1.87 + 3.24i)17-s + (3.36 − 2.77i)19-s + (2.12 − 3.68i)21-s + (0.208 + 0.361i)23-s + (−0.499 − 0.866i)25-s + 5.63·27-s + (4.85 + 8.40i)29-s − 4.93·31-s + ⋯ |
L(s) = 1 | + (0.430 − 0.745i)3-s + (−0.223 + 0.387i)5-s + 1.07·7-s + (0.129 + 0.223i)9-s + 0.260·11-s + (−0.0892 − 0.154i)13-s + (0.192 + 0.333i)15-s + (−0.453 + 0.785i)17-s + (0.770 − 0.636i)19-s + (0.463 − 0.803i)21-s + (0.0435 + 0.0753i)23-s + (−0.0999 − 0.173i)25-s + 1.08·27-s + (0.901 + 1.56i)29-s − 0.886·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.266855241\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.266855241\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.36 + 2.77i)T \) |
good | 3 | \( 1 + (-0.745 + 1.29i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 11 | \( 1 - 0.864T + 11T^{2} \) |
| 13 | \( 1 + (0.321 + 0.557i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.87 - 3.24i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.208 - 0.361i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.85 - 8.40i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4.93T + 31T^{2} \) |
| 37 | \( 1 - 6.36T + 37T^{2} \) |
| 41 | \( 1 + (-2.00 + 3.47i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.02 - 1.78i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.97 + 3.42i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.49 - 9.51i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.22 + 2.13i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.16 + 5.48i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.26 + 2.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.891 - 1.54i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.56 + 6.17i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.912 + 1.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.43T + 83T^{2} \) |
| 89 | \( 1 + (2.22 + 3.85i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.42 + 9.39i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.214353119792764857612028910926, −8.506109334995253785468837355620, −7.76535428393878696439868655285, −7.23411526454495927463055983930, −6.41511184656986168994918046172, −5.22786578404676555377855678069, −4.45745832090155217673250341569, −3.24765723229919047175921915060, −2.18170088007259169205682767218, −1.24939105551554254026306767424,
1.05840784890103537444358374895, 2.43889713321150768884480966694, 3.66719494208723822687389965638, 4.43988219131818128848311646465, 5.04603354899113457312183486949, 6.15599189315537980294342986898, 7.25985611698250047217014223258, 8.042623029889730704494256164711, 8.717638577390089465088732756476, 9.528084641297546727285952759098