| L(s) = 1 | − 2.56·5-s − 2.56·7-s + 1.43·11-s − 5.12·13-s + 5.68·17-s − 19-s + 0.876·23-s + 1.56·25-s − 8.24·29-s + 2·31-s + 6.56·35-s − 8·37-s − 3.12·41-s − 2.56·43-s + 5.68·47-s − 0.438·49-s + 12.2·53-s − 3.68·55-s + 12·59-s − 5.68·61-s + 13.1·65-s + 10.2·67-s + 11.9·73-s − 3.68·77-s + 13.3·79-s + 4·83-s − 14.5·85-s + ⋯ |
| L(s) = 1 | − 1.14·5-s − 0.968·7-s + 0.433·11-s − 1.42·13-s + 1.37·17-s − 0.229·19-s + 0.182·23-s + 0.312·25-s − 1.53·29-s + 0.359·31-s + 1.10·35-s − 1.31·37-s − 0.487·41-s − 0.390·43-s + 0.829·47-s − 0.0626·49-s + 1.68·53-s − 0.496·55-s + 1.56·59-s − 0.727·61-s + 1.62·65-s + 1.25·67-s + 1.39·73-s − 0.419·77-s + 1.50·79-s + 0.439·83-s − 1.57·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8737739233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8737739233\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 23 | \( 1 - 0.876T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 + 2.56T + 43T^{2} \) |
| 47 | \( 1 - 5.68T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.811271956755484105735883188250, −7.946844000161093680133625180627, −7.29230838927439506773497904809, −6.78861536407159452474600459067, −5.67692009011311142564267404269, −4.92451338795402021498930386138, −3.77034792848932059167411299865, −3.42816083367264814267277313218, −2.21162466423133495995436614675, −0.55791708194519210209535781600,
0.55791708194519210209535781600, 2.21162466423133495995436614675, 3.42816083367264814267277313218, 3.77034792848932059167411299865, 4.92451338795402021498930386138, 5.67692009011311142564267404269, 6.78861536407159452474600459067, 7.29230838927439506773497904809, 7.946844000161093680133625180627, 8.811271956755484105735883188250
