L(s) = 1 | − 1.41·5-s + 4i·13-s − 7.07i·17-s − 2.99·25-s + 9.89·29-s + 2i·37-s − 1.41i·41-s + 7·49-s + 7.07·53-s + 10i·61-s − 5.65i·65-s + 16·73-s + 10.0i·85-s − 18.3i·89-s + 8·97-s + ⋯ |
L(s) = 1 | − 0.632·5-s + 1.10i·13-s − 1.71i·17-s − 0.599·25-s + 1.83·29-s + 0.328i·37-s − 0.220i·41-s + 49-s + 0.971·53-s + 1.28i·61-s − 0.701i·65-s + 1.87·73-s + 1.08i·85-s − 1.94i·89-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.460374532\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.460374532\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 7.07iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.89T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 18.3iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.949712389346894551387562990251, −8.265858441063762168283622890958, −7.30104472235855271871575834171, −6.88597464229080004979746734613, −5.87370620691922394286486382455, −4.82036357999504079955750270292, −4.25244563909104272867023623688, −3.19001521777350587892045683535, −2.22053370541612669684806996507, −0.73521124694009539896184133465,
0.828481486293165882755916428500, 2.23770284636684538140397493434, 3.38840430817003176055897176635, 4.05500617367976127953831810791, 5.05063554526830658049546561940, 5.94589794816519740810081678625, 6.64847779523459034489738145170, 7.73240943549872675217553095536, 8.155959386762043577117941409035, 8.825520654645285017316818733511