L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.866 − 0.5i)7-s + (0.866 + 0.5i)11-s + (0.5 + 0.866i)13-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s − 0.999i·35-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 + 0.5i)47-s + 0.999i·55-s + (−0.866 + 0.5i)59-s + (0.5 − 0.866i)61-s + (−0.499 + 0.866i)65-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.866 − 0.5i)7-s + (0.866 + 0.5i)11-s + (0.5 + 0.866i)13-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s − 0.999i·35-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 + 0.5i)47-s + 0.999i·55-s + (−0.866 + 0.5i)59-s + (0.5 − 0.866i)61-s + (−0.499 + 0.866i)65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.167352422\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167352422\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.543885514652041122002703582331, −9.175082324694697094749423906716, −7.914398135782110932434010822138, −7.03004020555584160085217191519, −6.45671470629239178836423714981, −5.93230029871973798035470081817, −4.46405782510757234638656158648, −3.73866777472437973757336750750, −2.74950149137168911424797330324, −1.56310958133187847681320844723,
0.987392758379767511786723518841, 2.38665520872311839944329547963, 3.48679898519357348431707504018, 4.38707309781617242343030375863, 5.68389307123091056654112544851, 5.90081304946985317127610828765, 6.87934197690069529202837490502, 8.034300385598556594363479692767, 8.777892950440157883771614432740, 9.271426803935161031373606424733