L(s) = 1 | + (−0.5 − 0.866i)3-s + (2 + 3.46i)5-s + 3·7-s + (−0.499 + 0.866i)9-s − 2·11-s + (3.5 − 6.06i)13-s + (1.99 − 3.46i)15-s + (4 + 1.73i)19-s + (−1.5 − 2.59i)21-s + (−2 + 3.46i)23-s + (−5.49 + 9.52i)25-s + 0.999·27-s + (−2 + 3.46i)29-s − 31-s + (1 + 1.73i)33-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.894 + 1.54i)5-s + 1.13·7-s + (−0.166 + 0.288i)9-s − 0.603·11-s + (0.970 − 1.68i)13-s + (0.516 − 0.894i)15-s + (0.917 + 0.397i)19-s + (−0.327 − 0.566i)21-s + (−0.417 + 0.722i)23-s + (−1.09 + 1.90i)25-s + 0.192·27-s + (−0.371 + 0.643i)29-s − 0.179·31-s + (0.174 + 0.301i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.86668 + 0.403440i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86668 + 0.403440i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
good | 5 | \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (-3.5 + 6.06i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + (2 + 3.46i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (9 - 15.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.38833755844761193642002767259, −9.530703018723140011124739836572, −8.039838639431400733268567641045, −7.74133044888942073961991935825, −6.70562649621187785759088760674, −5.69570490498245550248519710087, −5.36519544049168526288556332843, −3.51694651573981621049979091414, −2.59239206423703109140038121591, −1.42393030589094712394548124738,
1.13723291485547177989060323285, 2.16441811249907008458530896566, 4.16044528634375194574928402451, 4.72058951128827525089432405681, 5.50860994201189556907059494307, 6.29563231633722245276164898782, 7.72195119982792751824206238234, 8.626699401600791270101429444939, 9.141236924835822165362449789660, 9.866682639141238037181147536627