| L(s) = 1 | + 2.82·7-s + 4.89·11-s + 4.89·13-s − 3.46·17-s − 6.92·19-s + 4·23-s + 8.48·29-s + 6.92·31-s + 4.89·37-s + 5.65·41-s − 11.3·43-s + 4·47-s + 1.00·49-s − 3.46·53-s + 4.89·59-s − 6·61-s + 5.65·67-s − 9.79·71-s + 13.8·77-s − 6.92·79-s + 16·83-s + 13.8·91-s − 9.79·97-s − 8.48·101-s + 2.82·103-s + 8·107-s − 6·109-s + ⋯ |
| L(s) = 1 | + 1.06·7-s + 1.47·11-s + 1.35·13-s − 0.840·17-s − 1.58·19-s + 0.834·23-s + 1.57·29-s + 1.24·31-s + 0.805·37-s + 0.883·41-s − 1.72·43-s + 0.583·47-s + 0.142·49-s − 0.475·53-s + 0.637·59-s − 0.768·61-s + 0.691·67-s − 1.16·71-s + 1.57·77-s − 0.779·79-s + 1.75·83-s + 1.45·91-s − 0.994·97-s − 0.844·101-s + 0.278·103-s + 0.773·107-s − 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.916910357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.916910357\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 9.79T + 97T^{2} \) |
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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
−8.253894317481687981833080861025, −7.10298849507247229663240671591, −6.33316580928714810731575860570, −6.19377139623520678022998168565, −4.81315907381007170632136467455, −4.43773130719410829931055272783, −3.72860764510698752871420361730, −2.64376317090830065079976792822, −1.64265355446194383011133989685, −0.957661388467314621964902275659,
0.957661388467314621964902275659, 1.64265355446194383011133989685, 2.64376317090830065079976792822, 3.72860764510698752871420361730, 4.43773130719410829931055272783, 4.81315907381007170632136467455, 6.19377139623520678022998168565, 6.33316580928714810731575860570, 7.10298849507247229663240671591, 8.253894317481687981833080861025
