L(s) = 1 | + (0.866 + 1.5i)3-s + (−1 + 1.73i)9-s + i·11-s + (−1 − 1.73i)17-s + (0.866 + 0.5i)19-s + (−0.5 + 0.866i)25-s − 1.73·27-s + (−1.5 + 0.866i)33-s + (1.5 − 0.866i)41-s + (1.73 − i)43-s − 49-s + (1.73 − 3i)51-s + 1.73i·57-s + (−0.866 − 1.5i)59-s + (−0.866 + 1.5i)67-s + ⋯ |
L(s) = 1 | + (0.866 + 1.5i)3-s + (−1 + 1.73i)9-s + i·11-s + (−1 − 1.73i)17-s + (0.866 + 0.5i)19-s + (−0.5 + 0.866i)25-s − 1.73·27-s + (−1.5 + 0.866i)33-s + (1.5 − 0.866i)41-s + (1.73 − i)43-s − 49-s + (1.73 − 3i)51-s + 1.73i·57-s + (−0.866 − 1.5i)59-s + (−0.866 + 1.5i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.340871427\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.340871427\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
good | 3 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
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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
−9.859825295837904118138054437879, −9.391416534658203502892946091109, −8.922177035390530661016816947780, −7.73307144997249478749434301097, −7.15159434707151250042789265430, −5.63838339759436784657694854294, −4.80308151849610231064714743141, −4.14816371993926317542914870285, −3.14486198115345709746484606932, −2.21709606127198372665030073163,
1.16938147683957324175620717374, 2.33145767731444444691075459073, 3.20312060153324186331548047291, 4.34459692288072221695634179016, 6.07511311539990027229873017074, 6.24973198387071939405427140802, 7.46563667873310485974817351383, 7.989860403924489491363240210229, 8.725386007813381043532035671217, 9.335797461812900121233535170021