| L(s) = 1 | − 3.46·5-s + 2.82·7-s + 2.44·11-s − 3.46·13-s − 4·17-s + 2.44·19-s − 2.82·23-s + 6.99·25-s − 3.46·29-s + 5.65·31-s − 9.79·35-s − 3.46·37-s + 2·41-s + 12.2·43-s + 11.3·47-s + 1.00·49-s − 10.3·53-s − 8.48·55-s − 2.44·59-s − 3.46·61-s + 11.9·65-s − 7.34·67-s + 2.82·71-s + 8·73-s + 6.92·77-s + 5.65·79-s − 7.34·83-s + ⋯ |
| L(s) = 1 | − 1.54·5-s + 1.06·7-s + 0.738·11-s − 0.960·13-s − 0.970·17-s + 0.561·19-s − 0.589·23-s + 1.39·25-s − 0.643·29-s + 1.01·31-s − 1.65·35-s − 0.569·37-s + 0.312·41-s + 1.86·43-s + 1.65·47-s + 0.142·49-s − 1.42·53-s − 1.14·55-s − 0.318·59-s − 0.443·61-s + 1.48·65-s − 0.897·67-s + 0.335·71-s + 0.936·73-s + 0.789·77-s + 0.636·79-s − 0.806·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.338910193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.338910193\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 3.46T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 + 7.34T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 + 7.34T + 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−8.130428461184891489514157176292, −7.64099961954276100417035159791, −7.15164670066556195703522091793, −6.21683234598726613286225659036, −5.15483387734794480295160065259, −4.39247592532531384138454415787, −4.06101778948021643459143981507, −2.96208868974482419389823342225, −1.90265539323342566053244812801, −0.64271795145394698784317487033,
0.64271795145394698784317487033, 1.90265539323342566053244812801, 2.96208868974482419389823342225, 4.06101778948021643459143981507, 4.39247592532531384138454415787, 5.15483387734794480295160065259, 6.21683234598726613286225659036, 7.15164670066556195703522091793, 7.64099961954276100417035159791, 8.130428461184891489514157176292
