L(s) = 1 | − 2-s + 1.41i·5-s + 7-s + 8-s − 1.41i·10-s − 1.41i·13-s − 14-s − 16-s + 17-s + 1.41i·19-s − 23-s − 1.00·25-s + 1.41i·26-s + 31-s − 34-s + 1.41i·35-s + ⋯ |
L(s) = 1 | − 2-s + 1.41i·5-s + 7-s + 8-s − 1.41i·10-s − 1.41i·13-s − 14-s − 16-s + 17-s + 1.41i·19-s − 23-s − 1.00·25-s + 1.41i·26-s + 31-s − 34-s + 1.41i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6344526683\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6344526683\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 109 | \( 1 - T \) |
good | 2 | \( 1 + T + T^{2} \) |
| 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.41iT - T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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
−10.24501922828186046826561671288, −9.829414406201761837786491810742, −8.348580867267987997166689650584, −7.985199993109958563126994420402, −7.36933289745844762765373199299, −6.17846975473152594450314417416, −5.24561599578540585682920266289, −3.96012904124212978167486029801, −2.84981227279322980961388545654, −1.46120439251126574485432159724,
1.02412040315215052849833468477, 2.02554769498266387721276537475, 4.21736695278699242965459839771, 4.67365549191244132068222878641, 5.60576794923655271891898181577, 7.06457925074114015450698880255, 7.88352989881114887565384654538, 8.603396486529950701470435105101, 9.089094810262748148313373468030, 9.762363584280643289465435771192