L(s) = 1 | + (0.707 + 0.707i)3-s + (2.01 − 1.71i)7-s + 1.00i·9-s − 4.27·11-s + (−2.31 − 2.31i)13-s + (−1.41 + 1.41i)17-s − 6.50·19-s + (2.63 + 0.209i)21-s + (−3.37 + 3.37i)23-s + (−0.707 + 0.707i)27-s − 8.27i·29-s − 3.04i·31-s + (−3.02 − 3.02i)33-s + (−7.68 − 7.68i)37-s − 3.27i·39-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (0.760 − 0.648i)7-s + 0.333i·9-s − 1.28·11-s + (−0.642 − 0.642i)13-s + (−0.342 + 0.342i)17-s − 1.49·19-s + (0.575 + 0.0456i)21-s + (−0.704 + 0.704i)23-s + (−0.136 + 0.136i)27-s − 1.53i·29-s − 0.546i·31-s + (−0.526 − 0.526i)33-s + (−1.26 − 1.26i)37-s − 0.524i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2245279007\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2245279007\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.01 + 1.71i)T \) |
good | 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 + (2.31 + 2.31i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.41 - 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 6.50T + 19T^{2} \) |
| 23 | \( 1 + (3.37 - 3.37i)T - 23iT^{2} \) |
| 29 | \( 1 + 8.27iT - 29T^{2} \) |
| 31 | \( 1 + 3.04iT - 31T^{2} \) |
| 37 | \( 1 + (7.68 + 7.68i)T + 37iT^{2} \) |
| 41 | \( 1 - 12.1iT - 41T^{2} \) |
| 43 | \( 1 + (5.53 - 5.53i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.61 - 8.61i)T - 53iT^{2} \) |
| 59 | \( 1 - 8.71T + 59T^{2} \) |
| 61 | \( 1 - 3.04iT - 61T^{2} \) |
| 67 | \( 1 + (-3.08 - 3.08i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.72T + 71T^{2} \) |
| 73 | \( 1 + (-7.07 - 7.07i)T + 73iT^{2} \) |
| 79 | \( 1 + 10.8iT - 79T^{2} \) |
| 83 | \( 1 + (-1.02 - 1.02i)T + 83iT^{2} \) |
| 89 | \( 1 - 7.88T + 89T^{2} \) |
| 97 | \( 1 + (1.92 - 1.92i)T - 97iT^{2} \) |
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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
−8.556608257814320889639548745607, −7.946523898189555630182803587402, −7.58426339896913419539913462117, −6.36964826048754349309118057115, −5.41975477518421612823042677399, −4.61428161953279423618952585887, −3.95249300876397559785982635865, −2.73864873942906960260320753165, −1.89464923135733285121096826371, −0.06559728315730933449873521817,
1.92624908155982278407040483188, 2.36209057675464684431244207196, 3.58726664277372585646484613806, 4.87191957637967586304985552001, 5.22108658813957561526100207293, 6.53367415319572779859738564543, 7.06403768352783486511854866916, 8.157052936840366680841744611625, 8.473979163863793144018793455492, 9.212029423153694639369312694158