| L(s) = 1 | + i·5-s − 2.71i·7-s − 1.37i·11-s + (−2.37 + 2.71i)13-s + 3.72·17-s + 1.01i·19-s + 1.70·23-s − 25-s + 5.43·29-s − 1.01i·31-s + 2.71·35-s − 3.72i·37-s − 1.37i·41-s − 4·43-s − 8.74i·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 1.02i·7-s − 0.413i·11-s + (−0.657 + 0.753i)13-s + 0.903·17-s + 0.231i·19-s + 0.355·23-s − 0.200·25-s + 1.00·29-s − 0.181i·31-s + 0.458·35-s − 0.612i·37-s − 0.214i·41-s − 0.609·43-s − 1.27i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.668409481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.668409481\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (2.37 - 2.71i)T \) |
| good | 7 | \( 1 + 2.71iT - 7T^{2} \) |
| 11 | \( 1 + 1.37iT - 11T^{2} \) |
| 17 | \( 1 - 3.72T + 17T^{2} \) |
| 19 | \( 1 - 1.01iT - 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 + 1.01iT - 31T^{2} \) |
| 37 | \( 1 + 3.72iT - 37T^{2} \) |
| 41 | \( 1 + 1.37iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 8.74iT - 47T^{2} \) |
| 53 | \( 1 - 9.15T + 53T^{2} \) |
| 59 | \( 1 + 8.74iT - 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 + 6.44iT - 67T^{2} \) |
| 71 | \( 1 + 4.11iT - 71T^{2} \) |
| 73 | \( 1 - 7.45iT - 73T^{2} \) |
| 79 | \( 1 + 3.37T + 79T^{2} \) |
| 83 | \( 1 + 11.4iT - 83T^{2} \) |
| 89 | \( 1 + 1.37iT - 89T^{2} \) |
| 97 | \( 1 + 1.70iT - 97T^{2} \) |
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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
−8.870992934085329184089235378113, −8.057677415712589478445618791412, −7.22362046016420802920212001361, −6.80316120169490826136936484560, −5.80206372806869532984961977230, −4.87465874621663746031700548293, −3.95687673747854928920085004479, −3.21684435202438720353012534680, −2.03521641045187954345658986660, −0.68563135126300476563900286382,
1.05601067177133197665713024846, 2.39666905727654047020594597739, 3.13347438714931250870554881824, 4.41759463295447627730458600311, 5.21888356041998209606187057461, 5.75234723262347787038328654350, 6.77672873756867793105165430365, 7.64312244717814802722044497571, 8.376346684738839929669736922378, 8.997993624938923751208765755283
