L(s) = 1 | + (0.0831 − 1.73i)3-s + 5-s + 4.02i·7-s + (−2.98 − 0.287i)9-s − 0.641·11-s − 2.89·13-s + (0.0831 − 1.73i)15-s + 3.10·17-s + 1.33i·19-s + (6.95 + 0.334i)21-s + (3.05 + 3.69i)23-s + 25-s + (−0.746 + 5.14i)27-s + 3.14i·29-s + 6.11·31-s + ⋯ |
L(s) = 1 | + (0.0480 − 0.998i)3-s + 0.447·5-s + 1.51i·7-s + (−0.995 − 0.0959i)9-s − 0.193·11-s − 0.804·13-s + (0.0214 − 0.446i)15-s + 0.752·17-s + 0.306i·19-s + (1.51 + 0.0729i)21-s + (0.636 + 0.771i)23-s + 0.200·25-s + (−0.143 + 0.989i)27-s + 0.583i·29-s + 1.09·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.529643524\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.529643524\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0831 + 1.73i)T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + (-3.05 - 3.69i)T \) |
good | 7 | \( 1 - 4.02iT - 7T^{2} \) |
| 11 | \( 1 + 0.641T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 - 1.33iT - 19T^{2} \) |
| 29 | \( 1 - 3.14iT - 29T^{2} \) |
| 31 | \( 1 - 6.11T + 31T^{2} \) |
| 37 | \( 1 - 7.06iT - 37T^{2} \) |
| 41 | \( 1 - 3.72iT - 41T^{2} \) |
| 43 | \( 1 - 1.20iT - 43T^{2} \) |
| 47 | \( 1 + 3.57iT - 47T^{2} \) |
| 53 | \( 1 - 5.12T + 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 0.921iT - 61T^{2} \) |
| 67 | \( 1 + 1.27iT - 67T^{2} \) |
| 71 | \( 1 + 1.44iT - 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 3.08iT - 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 - 13.1iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.515198925905106257887951965166, −8.777545952485105241125170991168, −8.081193500909035813763254277182, −7.23525659067549725919087178353, −6.31640265934313971935473165591, −5.60321957206022090834311069873, −4.97218852268920953650962087404, −3.11724782392929530097071625159, −2.47537043937397495535208656886, −1.38076995881213313114552729180,
0.64724620232520265309531002584, 2.46726537295204734987597755043, 3.52578975940688864041215175835, 4.43325718649041947911500106689, 5.07808447615038734395815432465, 6.12862973708058565898273996396, 7.13372791438042977747296195396, 7.86503108895860049532382643976, 8.861223149632572681402769068364, 9.727341627199429020505595533521