L(s) = 1 | − 2.65·5-s + 4.61·7-s + 4.21·11-s − 0.268·13-s + 0.800·17-s + 5.45·19-s − 4.25·23-s + 2.04·25-s − 8.09·29-s + 2.42·31-s − 12.2·35-s − 12.1·37-s + 4.26·41-s + 10.7·43-s + 6.40·47-s + 14.2·49-s + 4.79·53-s − 11.1·55-s + 0.639·59-s + 8.58·61-s + 0.713·65-s + 4.90·67-s + 15.9·71-s − 9.30·73-s + 19.4·77-s − 4.29·79-s − 13.9·83-s + ⋯ |
L(s) = 1 | − 1.18·5-s + 1.74·7-s + 1.27·11-s − 0.0745·13-s + 0.194·17-s + 1.25·19-s − 0.887·23-s + 0.408·25-s − 1.50·29-s + 0.435·31-s − 2.06·35-s − 1.99·37-s + 0.666·41-s + 1.63·43-s + 0.934·47-s + 2.03·49-s + 0.658·53-s − 1.50·55-s + 0.0832·59-s + 1.09·61-s + 0.0884·65-s + 0.598·67-s + 1.88·71-s − 1.08·73-s + 2.21·77-s − 0.482·79-s − 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.877710404\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.877710404\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.65T + 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 - 4.21T + 11T^{2} \) |
| 13 | \( 1 + 0.268T + 13T^{2} \) |
| 17 | \( 1 - 0.800T + 17T^{2} \) |
| 19 | \( 1 - 5.45T + 19T^{2} \) |
| 23 | \( 1 + 4.25T + 23T^{2} \) |
| 29 | \( 1 + 8.09T + 29T^{2} \) |
| 31 | \( 1 - 2.42T + 31T^{2} \) |
| 37 | \( 1 + 12.1T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 6.40T + 47T^{2} \) |
| 53 | \( 1 - 4.79T + 53T^{2} \) |
| 59 | \( 1 - 0.639T + 59T^{2} \) |
| 61 | \( 1 - 8.58T + 61T^{2} \) |
| 67 | \( 1 - 4.90T + 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 + 9.30T + 73T^{2} \) |
| 79 | \( 1 + 4.29T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 7.58T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.953630787072494051398899183789, −8.390260105168801720852951519729, −7.46347870421710130464270631817, −7.31408109400793368828252247042, −5.86655161030227196295595807008, −5.06661588519543943033287921708, −4.10836032295034478467192094842, −3.66705918972344048133266624304, −2.05942446517854358487533057086, −0.987116245044060137500236898989,
0.987116245044060137500236898989, 2.05942446517854358487533057086, 3.66705918972344048133266624304, 4.10836032295034478467192094842, 5.06661588519543943033287921708, 5.86655161030227196295595807008, 7.31408109400793368828252247042, 7.46347870421710130464270631817, 8.390260105168801720852951519729, 8.953630787072494051398899183789