L(s) = 1 | + (0.918 − 1.07i)2-s + 1.19·3-s + (−0.313 − 1.97i)4-s + (2.22 − 0.234i)5-s + (1.09 − 1.28i)6-s + (2.11 + 1.58i)7-s + (−2.41 − 1.47i)8-s − 1.58·9-s + (1.78 − 2.60i)10-s − 3.65·11-s + (−0.373 − 2.35i)12-s + 5.56i·13-s + (3.65 − 0.822i)14-s + (2.64 − 0.279i)15-s + (−3.80 + 1.23i)16-s − 0.808·17-s + ⋯ |
L(s) = 1 | + (0.649 − 0.760i)2-s + 0.687·3-s + (−0.156 − 0.987i)4-s + (0.994 − 0.105i)5-s + (0.446 − 0.522i)6-s + (0.800 + 0.599i)7-s + (−0.852 − 0.521i)8-s − 0.527·9-s + (0.565 − 0.824i)10-s − 1.10·11-s + (−0.107 − 0.678i)12-s + 1.54i·13-s + (0.975 − 0.219i)14-s + (0.683 − 0.0722i)15-s + (−0.950 + 0.309i)16-s − 0.196·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.96236 - 1.18498i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96236 - 1.18498i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.918 + 1.07i)T \) |
| 5 | \( 1 + (-2.22 + 0.234i)T \) |
| 7 | \( 1 + (-2.11 - 1.58i)T \) |
good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 - 5.56iT - 13T^{2} \) |
| 17 | \( 1 + 0.808T + 17T^{2} \) |
| 19 | \( 1 + 4.54iT - 19T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 + 8.36iT - 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 37 | \( 1 - 6.15T + 37T^{2} \) |
| 41 | \( 1 - 7.65iT - 41T^{2} \) |
| 43 | \( 1 - 2.66iT - 43T^{2} \) |
| 47 | \( 1 - 4.79iT - 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 + 8.92iT - 59T^{2} \) |
| 61 | \( 1 - 4.08T + 61T^{2} \) |
| 67 | \( 1 + 11.7iT - 67T^{2} \) |
| 71 | \( 1 - 8.17iT - 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 2.49iT - 79T^{2} \) |
| 83 | \( 1 + 5.75T + 83T^{2} \) |
| 89 | \( 1 + 4.95iT - 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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
−11.53966050455253723268909370506, −11.06073504150576349475161359208, −9.662495311972473204173063875598, −9.159358195638379504639415753082, −8.091696516395647578038405283161, −6.40300438914401024401477510480, −5.40374399070557772453991866101, −4.45120015402628611307855290195, −2.66148435892495884495161657616, −2.03539207384248062535903835962,
2.41285100968228858328816219387, 3.57319738579144397240871742830, 5.24351852569206136060500799319, 5.70992692936440441517345519532, 7.27501407795126044980089563061, 8.054147900754362462779871014784, 8.783477899686610681214982513366, 10.18381896712830813012867019615, 10.96682738886661355779358425211, 12.45886147424251506519974963275