| L(s) = 1 | − 2.76·5-s + 0.618·7-s − 5.51i·11-s − 2.58i·13-s + 5.12·17-s − 4.09·19-s + 5.83·23-s + 2.64·25-s − 0.795i·29-s + 4.82·31-s − 1.70·35-s + 1.19i·37-s + 8.20·41-s + 1.84i·43-s − 1.33i·47-s + ⋯ |
| L(s) = 1 | − 1.23·5-s + 0.233·7-s − 1.66i·11-s − 0.718i·13-s + 1.24·17-s − 0.940·19-s + 1.21·23-s + 0.528·25-s − 0.147i·29-s + 0.867·31-s − 0.288·35-s + 0.196i·37-s + 1.28·41-s + 0.281i·43-s − 0.194i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.126926446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.126926446\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (4.30 + 12.1i)T \) |
| good | 5 | \( 1 + 2.76T + 5T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + 5.51iT - 11T^{2} \) |
| 13 | \( 1 + 2.58iT - 13T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 + 4.09T + 19T^{2} \) |
| 23 | \( 1 - 5.83T + 23T^{2} \) |
| 29 | \( 1 + 0.795iT - 29T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 - 1.19iT - 37T^{2} \) |
| 41 | \( 1 - 8.20T + 41T^{2} \) |
| 43 | \( 1 - 1.84iT - 43T^{2} \) |
| 47 | \( 1 + 1.33iT - 47T^{2} \) |
| 53 | \( 1 - 4.14T + 53T^{2} \) |
| 59 | \( 1 - 4.91T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 + 1.39iT - 67T^{2} \) |
| 71 | \( 1 + 2.37T + 71T^{2} \) |
| 73 | \( 1 - 5.69iT - 73T^{2} \) |
| 79 | \( 1 + 7.74iT - 79T^{2} \) |
| 83 | \( 1 + 0.0266T + 83T^{2} \) |
| 89 | \( 1 + 8.65iT - 89T^{2} \) |
| 97 | \( 1 + 5.26T + 97T^{2} \) |
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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
−7.890829470796050738041604118590, −7.36662245844501622588602528265, −6.32849454414359180438200268244, −5.74535580666011767949825111805, −4.89985956175381655021342873167, −4.08486662429673418344601209218, −3.29163366832711443996644704209, −2.83202073454742542613937895465, −1.17423401765492337061006190449, −0.35763236454849786785304998287,
1.12054844195775283743767938811, 2.17931468552634667415432873372, 3.16840185413784397929000112463, 4.14627588014026009086250329755, 4.50230538757286457870608477160, 5.27625222863227009782241060339, 6.38894892424441853386234743658, 7.08160671133923074279979182732, 7.60833394762539075903555445470, 8.132082629116664245328923229865
