| L(s) = 1 | + 5-s − 4.77·7-s − 3.30·11-s + 3.19·13-s + 2.41·17-s + 2.24·19-s − 23-s + 25-s + 7.96·29-s − 10.3·31-s − 4.77·35-s − 11.2·37-s − 1.02·41-s + 10.0·43-s − 2.74·47-s + 15.8·49-s + 5.58·53-s − 3.30·55-s + 1.96·59-s − 8.24·61-s + 3.19·65-s − 7.71·67-s − 16.0·71-s − 7.92·73-s + 15.8·77-s + 12.5·83-s + 2.41·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.80·7-s − 0.997·11-s + 0.884·13-s + 0.585·17-s + 0.515·19-s − 0.208·23-s + 0.200·25-s + 1.47·29-s − 1.85·31-s − 0.807·35-s − 1.85·37-s − 0.160·41-s + 1.53·43-s − 0.400·47-s + 2.26·49-s + 0.767·53-s − 0.446·55-s + 0.256·59-s − 1.05·61-s + 0.395·65-s − 0.943·67-s − 1.90·71-s − 0.927·73-s + 1.80·77-s + 1.38·83-s + 0.261·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.333700313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.333700313\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 4.77T + 7T^{2} \) |
| 11 | \( 1 + 3.30T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 29 | \( 1 - 7.96T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 1.02T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 - 5.58T + 53T^{2} \) |
| 59 | \( 1 - 1.96T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + 7.71T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 7.92T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 3.05T + 89T^{2} \) |
| 97 | \( 1 + 5.43T + 97T^{2} \) |
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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
−7.62523578504340344127715946882, −7.12426793746826750400411119482, −6.32620635415832967358459785262, −5.79985145912396292439848601502, −5.26026551281995224423491400092, −4.13433753112427157699608144314, −3.24640860908209816468680139409, −2.94778673171720573651073140046, −1.79015294388898879380917452127, −0.55262943091967309427702018438,
0.55262943091967309427702018438, 1.79015294388898879380917452127, 2.94778673171720573651073140046, 3.24640860908209816468680139409, 4.13433753112427157699608144314, 5.26026551281995224423491400092, 5.79985145912396292439848601502, 6.32620635415832967358459785262, 7.12426793746826750400411119482, 7.62523578504340344127715946882
