L(s) = 1 | + (0.909 + 0.415i)2-s + (−0.755 − 0.654i)3-s + (0.654 + 0.755i)4-s + (0.841 − 0.540i)5-s + (−0.415 − 0.909i)6-s + (0.281 + 0.0405i)7-s + (0.281 + 0.959i)8-s + (0.142 + 0.989i)9-s + (0.989 − 0.142i)10-s − i·12-s + (0.239 + 0.153i)14-s + (−0.989 − 0.142i)15-s + (−0.142 + 0.989i)16-s + (−0.281 + 0.959i)18-s + (0.959 + 0.281i)20-s + (−0.186 − 0.215i)21-s + ⋯ |
L(s) = 1 | + (0.909 + 0.415i)2-s + (−0.755 − 0.654i)3-s + (0.654 + 0.755i)4-s + (0.841 − 0.540i)5-s + (−0.415 − 0.909i)6-s + (0.281 + 0.0405i)7-s + (0.281 + 0.959i)8-s + (0.142 + 0.989i)9-s + (0.989 − 0.142i)10-s − i·12-s + (0.239 + 0.153i)14-s + (−0.989 − 0.142i)15-s + (−0.142 + 0.989i)16-s + (−0.281 + 0.959i)18-s + (0.959 + 0.281i)20-s + (−0.186 − 0.215i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.718389375\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.718389375\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.909 - 0.415i)T \) |
| 3 | \( 1 + (0.755 + 0.654i)T \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (0.909 + 0.415i)T \) |
good | 7 | \( 1 + (-0.281 - 0.0405i)T + (0.959 + 0.281i)T^{2} \) |
| 11 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 13 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 17 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 + (-0.817 - 0.708i)T + (0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 37 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 41 | \( 1 + (1.07 + 1.66i)T + (-0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (0.368 - 1.25i)T + (-0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 + 1.68iT - T^{2} \) |
| 53 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 59 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.425 - 1.45i)T + (-0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (-1.74 - 0.797i)T + (0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 79 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (0.474 + 0.304i)T + (0.415 + 0.909i)T^{2} \) |
| 89 | \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (0.415 - 0.909i)T^{2} \) |
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
−10.01005792260087541038146873048, −8.626458567448142002957928335252, −8.110913424356366051459899119515, −6.96500880650334564033502213840, −6.47082583012625175374291259127, −5.49939527798756845679903306874, −5.11092847138498933198534954297, −4.08629948441472952210417322537, −2.55048973183302852950393541905, −1.59250029221569278667402238133,
1.55566671955738692515256605246, 2.82200918532926172209775096791, 3.81153612631053104412766138296, 4.78145578826814497602035280680, 5.47529623793782180217728793639, 6.29892471042314259186196039253, 6.77950238509338023225686102482, 8.093588162568468226301120419426, 9.625670561289702379719807792390, 9.789410509372589427144193172553