L(s) = 1 | + (0.885 − 0.464i)3-s + (−0.885 − 0.464i)4-s + (−0.616 − 1.17i)7-s + (0.568 − 0.822i)9-s − 12-s − 0.709·13-s + (0.568 + 0.822i)16-s + (1.32 + 1.17i)19-s + (−1.09 − 0.753i)21-s + (0.970 − 0.239i)25-s + (0.120 − 0.992i)27-s + 1.32i·28-s + (−0.688 + 1.81i)31-s + (−0.885 + 0.464i)36-s + (0.180 − 1.48i)37-s + ⋯ |
L(s) = 1 | + (0.885 − 0.464i)3-s + (−0.885 − 0.464i)4-s + (−0.616 − 1.17i)7-s + (0.568 − 0.822i)9-s − 12-s − 0.709·13-s + (0.568 + 0.822i)16-s + (1.32 + 1.17i)19-s + (−1.09 − 0.753i)21-s + (0.970 − 0.239i)25-s + (0.120 − 0.992i)27-s + 1.32i·28-s + (−0.688 + 1.81i)31-s + (−0.885 + 0.464i)36-s + (0.180 − 1.48i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8983174567\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8983174567\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.885 + 0.464i)T \) |
| 157 | \( 1 + (-0.885 + 0.464i)T \) |
good | 2 | \( 1 + (0.885 + 0.464i)T^{2} \) |
| 5 | \( 1 + (-0.970 + 0.239i)T^{2} \) |
| 7 | \( 1 + (0.616 + 1.17i)T + (-0.568 + 0.822i)T^{2} \) |
| 11 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 13 | \( 1 + 0.709T + T^{2} \) |
| 17 | \( 1 + (0.970 - 0.239i)T^{2} \) |
| 19 | \( 1 + (-1.32 - 1.17i)T + (0.120 + 0.992i)T^{2} \) |
| 23 | \( 1 + (-0.354 + 0.935i)T^{2} \) |
| 29 | \( 1 + (-0.970 - 0.239i)T^{2} \) |
| 31 | \( 1 + (0.688 - 1.81i)T + (-0.748 - 0.663i)T^{2} \) |
| 37 | \( 1 + (-0.180 + 1.48i)T + (-0.970 - 0.239i)T^{2} \) |
| 41 | \( 1 + (-0.354 - 0.935i)T^{2} \) |
| 43 | \( 1 + (0.764 - 1.45i)T + (-0.568 - 0.822i)T^{2} \) |
| 47 | \( 1 + (0.970 - 0.239i)T^{2} \) |
| 53 | \( 1 + (0.120 + 0.992i)T^{2} \) |
| 59 | \( 1 + (0.120 - 0.992i)T^{2} \) |
| 61 | \( 1 + (-0.317 + 0.358i)T + (-0.120 - 0.992i)T^{2} \) |
| 67 | \( 1 + (0.850 + 0.753i)T + (0.120 + 0.992i)T^{2} \) |
| 71 | \( 1 + (-0.120 - 0.992i)T^{2} \) |
| 73 | \( 1 + (-0.922 - 1.75i)T + (-0.568 + 0.822i)T^{2} \) |
| 79 | \( 1 + (-0.222 - 0.902i)T + (-0.885 + 0.464i)T^{2} \) |
| 83 | \( 1 + (-0.748 - 0.663i)T^{2} \) |
| 89 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 97 | \( 1 + (1.97 - 0.239i)T + (0.970 - 0.239i)T^{2} \) |
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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
−10.76068051183152275955271825872, −9.900107559604573398592804840863, −9.431758072680671891442979907228, −8.362033094285564922020425788887, −7.45683228720846276970652901319, −6.66035602996337025869791734528, −5.28412792198638399944547864074, −4.06317577187486867897551825372, −3.19243124714162449362933693156, −1.26210362603075135177926397394,
2.58585408423146059612916902917, 3.37143441410539665159214472636, 4.67754636980944872119214197033, 5.45828378212751013797492219106, 7.07685317373483924160806862161, 8.009219318425817066261872984333, 9.012474008153097586574203676021, 9.329908185636215792323434177545, 10.12115527889374716528882155171, 11.55181509433573355187943226367