L(s) = 1 | + (−1.67 + 1.67i)2-s + (2.55 − 1.56i)3-s − 1.58i·4-s + (−0.529 − 4.97i)5-s + (−1.66 + 6.89i)6-s + (6.78 − 1.73i)7-s + (−4.03 − 4.03i)8-s + (4.10 − 8.01i)9-s + (9.19 + 7.42i)10-s − 4.41i·11-s + (−2.48 − 4.06i)12-s + (1.62 + 1.62i)13-s + (−8.43 + 14.2i)14-s + (−9.13 − 11.8i)15-s + 19.8·16-s + (13.9 + 13.9i)17-s + ⋯ |
L(s) = 1 | + (−0.835 + 0.835i)2-s + (0.853 − 0.521i)3-s − 0.397i·4-s + (−0.105 − 0.994i)5-s + (−0.277 + 1.14i)6-s + (0.968 − 0.247i)7-s + (−0.503 − 0.503i)8-s + (0.455 − 0.890i)9-s + (0.919 + 0.742i)10-s − 0.401i·11-s + (−0.207 − 0.338i)12-s + (0.124 + 0.124i)13-s + (−0.602 + 1.01i)14-s + (−0.609 − 0.793i)15-s + 1.23·16-s + (0.819 + 0.819i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.19838 - 0.0670374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19838 - 0.0670374i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.55 + 1.56i)T \) |
| 5 | \( 1 + (0.529 + 4.97i)T \) |
| 7 | \( 1 + (-6.78 + 1.73i)T \) |
good | 2 | \( 1 + (1.67 - 1.67i)T - 4iT^{2} \) |
| 11 | \( 1 + 4.41iT - 121T^{2} \) |
| 13 | \( 1 + (-1.62 - 1.62i)T + 169iT^{2} \) |
| 17 | \( 1 + (-13.9 - 13.9i)T + 289iT^{2} \) |
| 19 | \( 1 + 0.694T + 361T^{2} \) |
| 23 | \( 1 + (-23.1 - 23.1i)T + 529iT^{2} \) |
| 29 | \( 1 + 49.1T + 841T^{2} \) |
| 31 | \( 1 + 33.8iT - 961T^{2} \) |
| 37 | \( 1 + (-2.02 - 2.02i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 32.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (30.4 - 30.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-18.7 - 18.7i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-33.8 - 33.8i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 23.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 12.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (56.3 + 56.3i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 92.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-95.4 - 95.4i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 100. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (5.62 - 5.62i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 158. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (37.1 - 37.1i)T - 9.40e3iT^{2} \) |
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
−13.52260243654036982978360035886, −12.67903690134871575157292181981, −11.49267494848932948448548708571, −9.691427625491853899718195968763, −8.784509847600853825377851488446, −8.039811405883521225685586396255, −7.33925497631310564993669323187, −5.70181434744947773107074882990, −3.80004238183120353878832100283, −1.28441409378062444644494788939,
2.03451511104998771911319784382, 3.28018073878603964144738963872, 5.16829208547516700250827565740, 7.29058226658591029604868413036, 8.382236658189556849253852072775, 9.375472595004189910510440930486, 10.39527508841324698387080266171, 11.02755002729796228911862702954, 12.09348712048292026840380435044, 13.77262140663627879987305600620