| L(s) = 1 | + 1.69·5-s − 4.12·7-s − 2.48·11-s + 1.35·13-s + 7.89·17-s + 5.08·19-s − 6.55·23-s − 2.13·25-s − 7.41·29-s + 4.69·31-s − 6.99·35-s − 0.0133·37-s + 1.02·41-s − 8.67·43-s − 5.31·47-s + 10.0·49-s − 12.1·53-s − 4.21·55-s + 0.0660·59-s + 10.3·61-s + 2.29·65-s + 6.10·67-s + 9.95·71-s − 3.76·73-s + 10.2·77-s + 11.4·79-s − 10.4·83-s + ⋯ |
| L(s) = 1 | + 0.757·5-s − 1.56·7-s − 0.750·11-s + 0.375·13-s + 1.91·17-s + 1.16·19-s − 1.36·23-s − 0.426·25-s − 1.37·29-s + 0.843·31-s − 1.18·35-s − 0.00219·37-s + 0.159·41-s − 1.32·43-s − 0.775·47-s + 1.43·49-s − 1.66·53-s − 0.568·55-s + 0.00860·59-s + 1.32·61-s + 0.284·65-s + 0.746·67-s + 1.18·71-s − 0.440·73-s + 1.17·77-s + 1.28·79-s − 1.14·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 1.69T + 5T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 7.89T + 17T^{2} \) |
| 19 | \( 1 - 5.08T + 19T^{2} \) |
| 23 | \( 1 + 6.55T + 23T^{2} \) |
| 29 | \( 1 + 7.41T + 29T^{2} \) |
| 31 | \( 1 - 4.69T + 31T^{2} \) |
| 37 | \( 1 + 0.0133T + 37T^{2} \) |
| 41 | \( 1 - 1.02T + 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 + 5.31T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 0.0660T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 6.10T + 67T^{2} \) |
| 71 | \( 1 - 9.95T + 71T^{2} \) |
| 73 | \( 1 + 3.76T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 6.46T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−7.80688980696895079088480514704, −6.94035275688156864876503711343, −6.15695090078023064798497641355, −5.69667607333770289638460357023, −5.11472174706567978677340231677, −3.68441649793508494055443195586, −3.34051342292403646661244352301, −2.42045187340929748678073705131, −1.32236105359509128837276337066, 0,
1.32236105359509128837276337066, 2.42045187340929748678073705131, 3.34051342292403646661244352301, 3.68441649793508494055443195586, 5.11472174706567978677340231677, 5.69667607333770289638460357023, 6.15695090078023064798497641355, 6.94035275688156864876503711343, 7.80688980696895079088480514704
