L(s) = 1 | + (−0.923 + 0.382i)7-s + (−0.923 − 0.382i)9-s + (−1.08 + 0.216i)11-s + (−0.707 + 1.70i)23-s + (0.382 + 0.923i)25-s + (−1.63 − 0.324i)29-s + (−1.08 + 1.63i)37-s + (−0.382 − 1.92i)43-s + (0.707 − 0.707i)49-s + (−0.382 + 0.0761i)53-s + 63-s + (0.0761 − 0.382i)67-s + (1.30 − 0.541i)71-s + (0.923 − 0.617i)77-s + (−0.541 + 0.541i)79-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.382i)7-s + (−0.923 − 0.382i)9-s + (−1.08 + 0.216i)11-s + (−0.707 + 1.70i)23-s + (0.382 + 0.923i)25-s + (−1.63 − 0.324i)29-s + (−1.08 + 1.63i)37-s + (−0.382 − 1.92i)43-s + (0.707 − 0.707i)49-s + (−0.382 + 0.0761i)53-s + 63-s + (0.0761 − 0.382i)67-s + (1.30 − 0.541i)71-s + (0.923 − 0.617i)77-s + (−0.541 + 0.541i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2657775540\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2657775540\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.923 - 0.382i)T \) |
good | 3 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 5 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 11 | \( 1 + (1.08 - 0.216i)T + (0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 23 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \) |
| 41 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.382 - 0.0761i)T + (0.923 - 0.382i)T^{2} \) |
| 59 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 61 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 + (-0.0761 + 0.382i)T + (-0.923 - 0.382i)T^{2} \) |
| 71 | \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 83 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 89 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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
−9.674211213120603516957327920290, −9.130078264270227908447396215671, −8.259947959604446210890888316030, −7.44652502559755005984112836181, −6.61613060772088626910788123295, −5.57220133757209465702592509776, −5.31277456088459691968956540809, −3.68730877566549624047420401110, −3.12242116436875192285188029185, −1.97821723851981163573984527693,
0.17913457193175150975863816429, 2.28594345944014048133979888164, 3.04477361268724157054064952584, 4.08806435357031403792119358873, 5.15906729238641378254182928519, 5.93519667461127539457901259727, 6.68290948455586779289843276474, 7.66385703763298711766563866123, 8.325144572860215569189556653220, 9.096752753424612116645721966419