L(s) = 1 | − 1.93i·5-s − i·7-s + 1.93·11-s − 0.517i·17-s + i·19-s + 1.41·23-s − 2.73·25-s − 1.93·35-s + 1.73i·43-s − 0.517·47-s − 3.73i·55-s − 1.73·61-s − 73-s − 1.93i·77-s − 1.41·83-s + ⋯ |
L(s) = 1 | − 1.93i·5-s − i·7-s + 1.93·11-s − 0.517i·17-s + i·19-s + 1.41·23-s − 2.73·25-s − 1.93·35-s + 1.73i·43-s − 0.517·47-s − 3.73i·55-s − 1.73·61-s − 73-s − 1.93i·77-s − 1.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.371927650\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.371927650\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 5 | \( 1 + 1.93iT - T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 - 1.93T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 0.517iT - T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - 1.73iT - T^{2} \) |
| 47 | \( 1 + 0.517T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.73T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.960991561834332072035619418378, −8.151452557727725765777327626004, −7.38220400086133185432927381928, −6.50396431876631373617483642110, −5.66357569012377386171569203779, −4.63612708856806230787812883548, −4.28689889387003170556112093130, −3.38758006108392029184637722084, −1.49902524997650440916984796045, −1.04593154159935357362566414735,
1.74651124885797434242918650109, 2.76052536516043770639961114238, 3.40057015642243860277110103333, 4.31823409474624584546809814276, 5.60448440565651029031673633335, 6.33563874156074736670490465420, 6.85981158370812464328063009971, 7.39492987757251544650102303416, 8.683018196585190271280468092495, 9.121707517024168119939807623808