L(s) = 1 | + (0.5 − 2.17i)5-s + (2.63 − 0.209i)7-s + (1.13 + 1.97i)11-s − 6.09i·13-s + (4.13 − 2.38i)17-s + (−2.13 + 3.70i)19-s + (0.774 + 0.447i)23-s + (−4.50 − 2.17i)25-s − 3.27·29-s + (2.13 + 3.70i)31-s + (0.862 − 5.85i)35-s + (−4.86 − 2.80i)37-s + 11.2·41-s − 6.50i·43-s + (−1.86 − 1.07i)47-s + ⋯ |
L(s) = 1 | + (0.223 − 0.974i)5-s + (0.996 − 0.0791i)7-s + (0.342 + 0.594i)11-s − 1.68i·13-s + (1.00 − 0.579i)17-s + (−0.490 + 0.849i)19-s + (0.161 + 0.0932i)23-s + (−0.900 − 0.435i)25-s − 0.608·29-s + (0.383 + 0.664i)31-s + (0.145 − 0.989i)35-s + (−0.799 − 0.461i)37-s + 1.76·41-s − 0.992i·43-s + (−0.271 − 0.156i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.930365858\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.930365858\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 2.17i)T \) |
| 7 | \( 1 + (-2.63 + 0.209i)T \) |
good | 11 | \( 1 + (-1.13 - 1.97i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6.09iT - 13T^{2} \) |
| 17 | \( 1 + (-4.13 + 2.38i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.13 - 3.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.774 - 0.447i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.27T + 29T^{2} \) |
| 31 | \( 1 + (-2.13 - 3.70i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.86 + 2.80i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 6.50iT - 43T^{2} \) |
| 47 | \( 1 + (1.86 + 1.07i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.41 - 3.70i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.13 + 3.70i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.774 - 1.34i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.0 + 6.95i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + (-1.86 + 1.07i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.137 - 0.238i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 5.67iT - 83T^{2} \) |
| 89 | \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−9.544708759503687109462939655702, −8.646781717839846553377872722512, −7.925548797912663070801221535702, −7.39018437433574627644986244999, −5.90529946929309367687627780349, −5.30863740790176949616251094760, −4.55027447178937623427004660504, −3.43764124208605952986862173797, −1.97673215619286033884015768245, −0.898795778203968080107793765437,
1.51373439891561765317885785346, 2.54293494173146121070692595021, 3.77134714037811099761921947160, 4.62384029694214488069549362008, 5.80753402185368277305551355514, 6.51873812299085196244697985307, 7.34392276777199339820933440705, 8.182412356822877777656559251577, 9.067911847754295896453718637143, 9.788158257262732225638644919990