L(s) = 1 | + (−0.866 − 0.5i)2-s + i·8-s + (0.5 − 0.866i)16-s + i·17-s + 19-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)34-s + (−0.866 − 0.5i)38-s + 0.999·46-s + (1.73 + i)47-s + (0.5 + 0.866i)49-s − i·53-s + (0.5 − 0.866i)61-s − 0.999i·62-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + i·8-s + (0.5 − 0.866i)16-s + i·17-s + 19-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)34-s + (−0.866 − 0.5i)38-s + 0.999·46-s + (1.73 + i)47-s + (0.5 + 0.866i)49-s − i·53-s + (0.5 − 0.866i)61-s − 0.999i·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6695687030\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6695687030\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + iT - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−9.449518229671297645416639030100, −8.606411855580924525124675433534, −8.023021476433413874483974429290, −7.20526376663583571598193066095, −6.06028525691658715527193952385, −5.41546441004578799070022287827, −4.38599537733489761626788181243, −3.28404949262021698157272904817, −2.15575717556613149128907223841, −1.13816103219755869098658394721,
0.796436407443435296753350459483, 2.42929120577766331274259350851, 3.57076606916853206557710788218, 4.45715624614404614809135921284, 5.53790045439090630797049024542, 6.41247542612574389861234128499, 7.32760914496690127088645365632, 7.69269835053270633829700349082, 8.659129656626940218986409012421, 9.177953539975310770505828187088