L(s) = 1 | − 3i·3-s + 13i·7-s − 9·9-s + 23i·13-s − 11·19-s + 39·21-s + 27i·27-s + 59·31-s − 26i·37-s + 69·39-s + 83i·43-s − 120·49-s + 33i·57-s − 121·61-s − 117i·63-s + ⋯ |
L(s) = 1 | − i·3-s + 1.85i·7-s − 9-s + 1.76i·13-s − 0.578·19-s + 1.85·21-s + i·27-s + 1.90·31-s − 0.702i·37-s + 1.76·39-s + 1.93i·43-s − 2.44·49-s + 0.578i·57-s − 1.98·61-s − 1.85i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05048 + 0.649234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05048 + 0.649234i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 13iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 23iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 + 11T + 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 59T + 961T^{2} \) |
| 37 | \( 1 + 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 83iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 121T + 3.72e3T^{2} \) |
| 67 | \( 1 - 13iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 142T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 167iT - 9.40e3T^{2} \) |
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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
−11.91893341119461219277812358815, −11.14313299260001934214916051504, −9.454965557907483102144459302987, −8.815381953638103562640972287861, −7.953258683561038491650617967762, −6.56719051155626428363894701120, −6.04468014680664510962976631743, −4.71065036191316452858260308615, −2.76878541219959199789426933710, −1.80714191230484323480230343056,
0.59979971103601221384346745151, 3.09567823811564449111867640423, 4.09292806883798939428159789023, 5.05874655275264717236550106787, 6.37137258114001689092462252030, 7.64802806328796646043794127308, 8.437165553783129398806432879654, 9.854263914352636454474409972376, 10.43376664125113216007951691107, 10.90882906484152064632781482593