| L(s) = 1 | − 3-s + 0.438·5-s + 2.51·7-s + 9-s − 4.03·11-s + 1.44·13-s − 0.438·15-s + 1.85·17-s + 0.369·19-s − 2.51·21-s − 9.02·23-s − 4.80·25-s − 27-s − 8.65·29-s − 6.50·31-s + 4.03·33-s + 1.10·35-s + 8.50·37-s − 1.44·39-s + 6.94·41-s + 6.70·43-s + 0.438·45-s + 9.14·47-s − 0.652·49-s − 1.85·51-s + 1.30·53-s − 1.76·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.196·5-s + 0.952·7-s + 0.333·9-s − 1.21·11-s + 0.400·13-s − 0.113·15-s + 0.449·17-s + 0.0846·19-s − 0.549·21-s − 1.88·23-s − 0.961·25-s − 0.192·27-s − 1.60·29-s − 1.16·31-s + 0.701·33-s + 0.186·35-s + 1.39·37-s − 0.231·39-s + 1.08·41-s + 1.02·43-s + 0.0653·45-s + 1.33·47-s − 0.0931·49-s − 0.259·51-s + 0.179·53-s − 0.238·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516726227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516726227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 0.438T + 5T^{2} \) |
| 7 | \( 1 - 2.51T + 7T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 - 0.369T + 19T^{2} \) |
| 23 | \( 1 + 9.02T + 23T^{2} \) |
| 29 | \( 1 + 8.65T + 29T^{2} \) |
| 31 | \( 1 + 6.50T + 31T^{2} \) |
| 37 | \( 1 - 8.50T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 - 6.70T + 43T^{2} \) |
| 47 | \( 1 - 9.14T + 47T^{2} \) |
| 53 | \( 1 - 1.30T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 + 0.264T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 0.243T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 4.31T + 79T^{2} \) |
| 83 | \( 1 - 6.84T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 2.37T + 97T^{2} \) |
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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
−7.72393683596740390468040868228, −7.43127105234583455543175320433, −6.20172532549943726370399365109, −5.60956057401802298849259617690, −5.33695503609097483459815613741, −4.22993726136073772223443905945, −3.77945260835696074803782839543, −2.38714589708734632940836959426, −1.86154212507835864955608984028, −0.61880431403807868338950344151,
0.61880431403807868338950344151, 1.86154212507835864955608984028, 2.38714589708734632940836959426, 3.77945260835696074803782839543, 4.22993726136073772223443905945, 5.33695503609097483459815613741, 5.60956057401802298849259617690, 6.20172532549943726370399365109, 7.43127105234583455543175320433, 7.72393683596740390468040868228
