| L(s) = 1 | − 3.30·5-s − 4.47·7-s − 1.74i·11-s − 2.94i·13-s + 2.90·17-s + 3.77·19-s + 0.201·23-s + 5.90·25-s + 6.23i·29-s + 4.31·31-s + 14.7·35-s − 5.99i·37-s + 6.64·41-s + 6.12i·43-s − 5.02i·47-s + ⋯ |
| L(s) = 1 | − 1.47·5-s − 1.69·7-s − 0.526i·11-s − 0.817i·13-s + 0.704·17-s + 0.865·19-s + 0.0419·23-s + 1.18·25-s + 1.15i·29-s + 0.775·31-s + 2.49·35-s − 0.985i·37-s + 1.03·41-s + 0.934i·43-s − 0.733i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3141020063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3141020063\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-9.47 - 8.78i)T \) |
| good | 5 | \( 1 + 3.30T + 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 + 1.74iT - 11T^{2} \) |
| 13 | \( 1 + 2.94iT - 13T^{2} \) |
| 17 | \( 1 - 2.90T + 17T^{2} \) |
| 19 | \( 1 - 3.77T + 19T^{2} \) |
| 23 | \( 1 - 0.201T + 23T^{2} \) |
| 29 | \( 1 - 6.23iT - 29T^{2} \) |
| 31 | \( 1 - 4.31T + 31T^{2} \) |
| 37 | \( 1 + 5.99iT - 37T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 - 6.12iT - 43T^{2} \) |
| 47 | \( 1 + 5.02iT - 47T^{2} \) |
| 53 | \( 1 + 2.29T + 53T^{2} \) |
| 59 | \( 1 + 0.540T + 59T^{2} \) |
| 61 | \( 1 + 8.87T + 61T^{2} \) |
| 67 | \( 1 + 12.8iT - 67T^{2} \) |
| 71 | \( 1 - 5.03T + 71T^{2} \) |
| 73 | \( 1 + 4.72iT - 73T^{2} \) |
| 79 | \( 1 + 0.282iT - 79T^{2} \) |
| 83 | \( 1 - 3.56T + 83T^{2} \) |
| 89 | \( 1 + 3.81iT - 89T^{2} \) |
| 97 | \( 1 + 8.03T + 97T^{2} \) |
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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
−7.69935934956716651583099564669, −7.18160666717551827083620703686, −6.35775999677766612659225479388, −5.70087759814528251850500615972, −4.80972083514347227004903617932, −3.75771813090928968668878479015, −3.32641071789423829675375445950, −2.82442931242154310582151320178, −0.948864558691070459057638171400, −0.12148958051405640143344863705,
0.998744052947588404729419545882, 2.59275564502671229198885400731, 3.28109081603648048366220931471, 3.98157688048029125265645784401, 4.53489753466954301062631062366, 5.65087814801959134990676027418, 6.45103871327579308347181929286, 7.04352307711726262747314292653, 7.63126567380155628091226657865, 8.266599379870594872711930424081
