| L(s) = 1 | + 5-s − 3·7-s + 4.72·11-s − 5·13-s + 5.72·17-s + 5.72·19-s + 23-s + 25-s − 1.72·29-s + 3·31-s − 3·35-s − 37-s − 5·41-s + 5.72·43-s − 3·47-s + 2·49-s + 4·53-s + 4.72·55-s − 3.27·59-s − 6.44·61-s − 5·65-s − 3.72·67-s − 5.72·71-s − 2.72·73-s − 14.1·77-s + 5.27·79-s − 14.7·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 1.42·11-s − 1.38·13-s + 1.38·17-s + 1.31·19-s + 0.208·23-s + 0.200·25-s − 0.319·29-s + 0.538·31-s − 0.507·35-s − 0.164·37-s − 0.780·41-s + 0.872·43-s − 0.437·47-s + 0.285·49-s + 0.549·53-s + 0.636·55-s − 0.426·59-s − 0.824·61-s − 0.620·65-s − 0.454·67-s − 0.678·71-s − 0.318·73-s − 1.61·77-s + 0.594·79-s − 1.61·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.044209728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.044209728\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 4.72T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 - 5.72T + 17T^{2} \) |
| 19 | \( 1 - 5.72T + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + 1.72T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 5.72T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 3.27T + 59T^{2} \) |
| 61 | \( 1 + 6.44T + 61T^{2} \) |
| 67 | \( 1 + 3.72T + 67T^{2} \) |
| 71 | \( 1 + 5.72T + 71T^{2} \) |
| 73 | \( 1 + 2.72T + 73T^{2} \) |
| 79 | \( 1 - 5.27T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.77775839641179227211346563356, −7.25782090450358922248232726912, −6.59343075325268909788644004323, −5.91851255587601462159152827097, −5.25268384064916944656695881266, −4.38749597730970959522392451548, −3.35539167245185551041680861047, −2.98759142307471205148437383865, −1.75801290051879270650805376716, −0.75262981294069285616655057676,
0.75262981294069285616655057676, 1.75801290051879270650805376716, 2.98759142307471205148437383865, 3.35539167245185551041680861047, 4.38749597730970959522392451548, 5.25268384064916944656695881266, 5.91851255587601462159152827097, 6.59343075325268909788644004323, 7.25782090450358922248232726912, 7.77775839641179227211346563356
