| L(s) = 1 | + i·3-s + (0.432 − 2.19i)5-s − 9-s + 2.86·11-s − i·13-s + (2.19 + 0.432i)15-s − 5.52i·17-s + 3.52·19-s − 7.52i·23-s + (−4.62 − 1.89i)25-s − i·27-s − 6.77·29-s − 5.72·31-s + 2.86i·33-s + 3.72i·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.193 − 0.981i)5-s − 0.333·9-s + 0.863·11-s − 0.277i·13-s + (0.566 + 0.111i)15-s − 1.33i·17-s + 0.808·19-s − 1.56i·23-s + (−0.925 − 0.379i)25-s − 0.192i·27-s − 1.25·29-s − 1.02·31-s + 0.498i·33-s + 0.613i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.421925546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421925546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-0.432 + 2.19i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 17 | \( 1 + 5.52iT - 17T^{2} \) |
| 19 | \( 1 - 3.52T + 19T^{2} \) |
| 23 | \( 1 + 7.52iT - 23T^{2} \) |
| 29 | \( 1 + 6.77T + 29T^{2} \) |
| 31 | \( 1 + 5.72T + 31T^{2} \) |
| 37 | \( 1 - 3.72iT - 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 5.52iT - 43T^{2} \) |
| 47 | \( 1 - 8.65iT - 47T^{2} \) |
| 53 | \( 1 + 6.77iT - 53T^{2} \) |
| 59 | \( 1 - 0.593T + 59T^{2} \) |
| 61 | \( 1 + 5.25T + 61T^{2} \) |
| 67 | \( 1 + 10.5iT - 67T^{2} \) |
| 71 | \( 1 - 2.38T + 71T^{2} \) |
| 73 | \( 1 - 5.45iT - 73T^{2} \) |
| 79 | \( 1 - 2.47T + 79T^{2} \) |
| 83 | \( 1 + 8.11iT - 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−8.653960438816911146143198554705, −7.86385346803128622375795995910, −6.98023794501156461579400304303, −6.08326113847345923609149590202, −5.22611423518205722776456488711, −4.71502003912366397323264245244, −3.84284258198109916668165331117, −2.90570967842718281498014123252, −1.64811885978877486401605361282, −0.43442245076966229069987308226,
1.47437469455835327633122598897, 2.14746807883972408889166258360, 3.55415593881964060690529633781, 3.77085828579888510112810546576, 5.42747770206425852654020393481, 5.87283293147336683891715076947, 6.82781925388679212569334095157, 7.24682617983848337553944760922, 7.974736553810754390217886047870, 9.002442362749210980285217041599
