L(s) = 1 | + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s + (−0.5 − 0.866i)14-s + 16-s + (−0.866 + 0.5i)17-s + (0.866 − 0.5i)19-s + (−0.499 + 0.866i)21-s + (0.866 + 0.5i)22-s + ⋯ |
L(s) = 1 | + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s + (−0.5 − 0.866i)14-s + 16-s + (−0.866 + 0.5i)17-s + (0.866 − 0.5i)19-s + (−0.499 + 0.866i)21-s + (0.866 + 0.5i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.145042282\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.145042282\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + iT \) |
good | 3 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.639100339679315018591954244564, −7.44762588821200449430172462702, −7.04978446595181953375101551858, −6.41309620907686691632061984662, −5.98849585522427375038919657611, −4.78094655509316772459830014847, −4.00874107000753962397629509879, −3.39232547889014650728683080049, −1.97577363131567864816377018114, −1.14953641591110374745511598523,
1.60398469939576053752727578108, 2.91152825212158226549359492547, 3.60409136017536410054609815505, 4.36133506507771311799667529868, 5.34045594637151400816262619322, 5.68847091455104255482246287730, 6.49654498296735004956818305263, 7.22984041900066411618286028149, 8.385452844529253053269588592912, 9.059201486478345986913352170924