| L(s) = 1 | + 0.473·5-s + 4.55·7-s − 3.49·11-s − 0.0840·13-s + 3.61·17-s + 3.61·19-s − 2.82·23-s − 4.77·25-s − 7.30·29-s − 0.557·31-s + 2.15·35-s + 6.20·37-s + 9.27·41-s + 2.27·43-s + 2.82·47-s + 13.7·49-s − 0.697·53-s − 1.65·55-s − 5.65·59-s − 3.85·61-s − 0.0397·65-s − 5.33·67-s + 9.11·71-s − 0.541·73-s − 15.9·77-s + 10.9·79-s + 15.0·83-s + ⋯ |
| L(s) = 1 | + 0.211·5-s + 1.72·7-s − 1.05·11-s − 0.0233·13-s + 0.877·17-s + 0.829·19-s − 0.589·23-s − 0.955·25-s − 1.35·29-s − 0.100·31-s + 0.364·35-s + 1.01·37-s + 1.44·41-s + 0.347·43-s + 0.412·47-s + 1.96·49-s − 0.0958·53-s − 0.223·55-s − 0.736·59-s − 0.494·61-s − 0.00493·65-s − 0.652·67-s + 1.08·71-s − 0.0633·73-s − 1.81·77-s + 1.23·79-s + 1.65·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.660214202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.660214202\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.473T + 5T^{2} \) |
| 7 | \( 1 - 4.55T + 7T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 + 0.0840T + 13T^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 + 0.557T + 31T^{2} \) |
| 37 | \( 1 - 6.20T + 37T^{2} \) |
| 41 | \( 1 - 9.27T + 41T^{2} \) |
| 43 | \( 1 - 2.27T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 0.697T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 + 5.33T + 67T^{2} \) |
| 71 | \( 1 - 9.11T + 71T^{2} \) |
| 73 | \( 1 + 0.541T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−7.74235739407611804067283340316, −7.48559345508220011856996872732, −6.11217120576067927219001726843, −5.57625191477833637471809439896, −5.05614398351850211144988490634, −4.32629267628356351082820007230, −3.49070344139592879068885906983, −2.41471782207082450786164746016, −1.81603226532849095313832267296, −0.804413752885254026448738341983,
0.804413752885254026448738341983, 1.81603226532849095313832267296, 2.41471782207082450786164746016, 3.49070344139592879068885906983, 4.32629267628356351082820007230, 5.05614398351850211144988490634, 5.57625191477833637471809439896, 6.11217120576067927219001726843, 7.48559345508220011856996872732, 7.74235739407611804067283340316
