| L(s) = 1 | + 1.93·3-s − 3.18·5-s + 4.42·7-s + 0.745·9-s + 13-s − 6.17·15-s + 1.49·17-s + 1.76·19-s + 8.56·21-s + 1.61·23-s + 5.17·25-s − 4.36·27-s + 9.36·29-s − 5.31·31-s − 14.1·35-s − 0.810·37-s + 1.93·39-s + 1.91·41-s − 5.23·43-s − 2.37·45-s + 8.37·47-s + 12.6·49-s + 2.88·51-s + 6.61·53-s + 3.41·57-s + 11.4·59-s − 12.2·61-s + ⋯ |
| L(s) = 1 | + 1.11·3-s − 1.42·5-s + 1.67·7-s + 0.248·9-s + 0.277·13-s − 1.59·15-s + 0.361·17-s + 0.404·19-s + 1.86·21-s + 0.337·23-s + 1.03·25-s − 0.839·27-s + 1.73·29-s − 0.955·31-s − 2.38·35-s − 0.133·37-s + 0.309·39-s + 0.299·41-s − 0.798·43-s − 0.354·45-s + 1.22·47-s + 1.80·49-s + 0.404·51-s + 0.908·53-s + 0.451·57-s + 1.48·59-s − 1.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.907608900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.907608900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.93T + 3T^{2} \) |
| 5 | \( 1 + 3.18T + 5T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 19 | \( 1 - 1.76T + 19T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 - 9.36T + 29T^{2} \) |
| 31 | \( 1 + 5.31T + 31T^{2} \) |
| 37 | \( 1 + 0.810T + 37T^{2} \) |
| 41 | \( 1 - 1.91T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 - 8.37T + 47T^{2} \) |
| 53 | \( 1 - 6.61T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 0.935T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 3.18T + 97T^{2} \) |
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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
−8.042534972963250442081104116431, −7.58774330398627344729902560641, −7.08182568868598298207656863658, −5.79663960336833790775948789143, −4.92630222079990071995890136688, −4.31187800175164790964154796495, −3.60535334259879105964097376041, −2.89028161637413489325706184178, −1.90899656978815358049731234413, −0.872144092076149211595445715507,
0.872144092076149211595445715507, 1.90899656978815358049731234413, 2.89028161637413489325706184178, 3.60535334259879105964097376041, 4.31187800175164790964154796495, 4.92630222079990071995890136688, 5.79663960336833790775948789143, 7.08182568868598298207656863658, 7.58774330398627344729902560641, 8.042534972963250442081104116431
