L(s) = 1 | + (0.258 + 0.965i)2-s + (0.258 − 0.965i)3-s + (−0.866 + 0.499i)4-s + 6-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−1.5 − 0.866i)11-s + (0.258 + 0.965i)12-s + (0.500 − 0.866i)16-s + (0.707 − 0.707i)17-s + (0.258 − 0.965i)18-s − 2i·19-s + (0.448 − 1.67i)22-s + (−0.866 + 0.5i)24-s + (−0.707 + 0.707i)27-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (0.258 − 0.965i)3-s + (−0.866 + 0.499i)4-s + 6-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−1.5 − 0.866i)11-s + (0.258 + 0.965i)12-s + (0.500 − 0.866i)16-s + (0.707 − 0.707i)17-s + (0.258 − 0.965i)18-s − 2i·19-s + (0.448 − 1.67i)22-s + (−0.866 + 0.5i)24-s + (−0.707 + 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9333247558\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9333247558\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 19 | \( 1 + 2iT - T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + 1.73T + T^{2} \) |
| 97 | \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \) |
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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
−8.987524896066500925783377855926, −8.324329209469722851273016462814, −7.63678263843518123324689056677, −7.09531276876579411350495386025, −6.20204284715359451763261878440, −5.44090625738376086517404025227, −4.73158218509166383123112270314, −3.21463747184106147804899495600, −2.65471491415426207051933902326, −0.61138714427806882444659921441,
1.84309416087299561997081507059, 2.81193297409781177621192577115, 3.74276156036104198144548799378, 4.42899211420397330370308116252, 5.44627272255301423756405452260, 5.81591339737399440000860977868, 7.58493557241131463451346516047, 8.199909121451316829877861190767, 9.024489606336043090030199032315, 9.968938736573414551638737167379