| L(s) = 1 | + 9.48·5-s + 3.48·7-s + 21.9·11-s + 64.9·25-s − 50·29-s − 23.4·31-s + 33.0·35-s − 36.8·49-s − 89.4·53-s + 208.·55-s − 10·59-s + 93.7·73-s + 76.5·77-s + 58·79-s − 151.·83-s + 61.0·97-s − 35.6·101-s + 10·103-s + 126.·107-s + ⋯ |
| L(s) = 1 | + 1.89·5-s + 0.497·7-s + 1.99·11-s + 2.59·25-s − 1.72·29-s − 0.755·31-s + 0.944·35-s − 0.752·49-s − 1.68·53-s + 3.78·55-s − 0.169·59-s + 1.28·73-s + 0.994·77-s + 0.734·79-s − 1.82·83-s + 0.629·97-s − 0.352·101-s + 0.0970·103-s + 1.18·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.184822618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.184822618\) |
| \(L(2)\) |
|
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 | \( 1 - 9.48T + 25T^{2} \) |
| 7 | \( 1 - 3.48T + 49T^{2} \) |
| 11 | \( 1 - 21.9T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 50T + 841T^{2} \) |
| 31 | \( 1 + 23.4T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 89.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 10T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 93.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58T + 6.24e3T^{2} \) |
| 83 | \( 1 + 151.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 61.0T + 9.40e3T^{2} \) |
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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
−9.560213335110203030830127974273, −9.460777383958015667077349693895, −8.552273429319934084031127820674, −7.19156969051759246235190534584, −6.35945967156758340862882185384, −5.73824118384244416846666549794, −4.74274250345545176808339209560, −3.51006313046375266989529034751, −2.00641460455644028005339769993, −1.37390433329730973153818697648,
1.37390433329730973153818697648, 2.00641460455644028005339769993, 3.51006313046375266989529034751, 4.74274250345545176808339209560, 5.73824118384244416846666549794, 6.35945967156758340862882185384, 7.19156969051759246235190534584, 8.552273429319934084031127820674, 9.460777383958015667077349693895, 9.560213335110203030830127974273
