| L(s) = 1 | − 2.55·3-s − 4.16·5-s − 2.08·7-s + 3.51·9-s + 1.07·11-s + 1.35·13-s + 10.6·15-s + 3.15·17-s + 1.85·19-s + 5.33·21-s − 3.48·23-s + 12.3·25-s − 1.32·27-s − 9.50·29-s + 8.19·31-s − 2.74·33-s + 8.70·35-s − 1.31·37-s − 3.46·39-s − 9.93·41-s − 8.76·43-s − 14.6·45-s − 10.8·47-s − 2.63·49-s − 8.06·51-s − 8.26·53-s − 4.47·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 1.86·5-s − 0.789·7-s + 1.17·9-s + 0.323·11-s + 0.376·13-s + 2.74·15-s + 0.766·17-s + 0.425·19-s + 1.16·21-s − 0.726·23-s + 2.47·25-s − 0.254·27-s − 1.76·29-s + 1.47·31-s − 0.477·33-s + 1.47·35-s − 0.216·37-s − 0.555·39-s − 1.55·41-s − 1.33·43-s − 2.18·45-s − 1.57·47-s − 0.376·49-s − 1.12·51-s − 1.13·53-s − 0.603·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2204171677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2204171677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 2.55T + 3T^{2} \) |
| 5 | \( 1 + 4.16T + 5T^{2} \) |
| 7 | \( 1 + 2.08T + 7T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 23 | \( 1 + 3.48T + 23T^{2} \) |
| 29 | \( 1 + 9.50T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 + 1.31T + 37T^{2} \) |
| 41 | \( 1 + 9.93T + 41T^{2} \) |
| 43 | \( 1 + 8.76T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 8.26T + 53T^{2} \) |
| 59 | \( 1 + 3.05T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 9.10T + 67T^{2} \) |
| 71 | \( 1 - 6.64T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 3.49T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.004129864743286247265536396536, −7.24600166645201440237598813970, −6.60287047070686334700606962975, −6.08760654546769336151711450798, −5.10549424287770269970810977104, −4.58230054934942135535698092533, −3.54106223174636414109128596201, −3.32091606695519037752022440180, −1.40916654658567091149213560332, −0.28435087282239037063862755972,
0.28435087282239037063862755972, 1.40916654658567091149213560332, 3.32091606695519037752022440180, 3.54106223174636414109128596201, 4.58230054934942135535698092533, 5.10549424287770269970810977104, 6.08760654546769336151711450798, 6.60287047070686334700606962975, 7.24600166645201440237598813970, 8.004129864743286247265536396536
