| L(s) = 1 | + 1.45·3-s + 1.40·5-s + 7-s − 0.895·9-s − 4.40·11-s + 2.03·15-s + 0.145·17-s + 1.99·19-s + 1.45·21-s + 6.06·23-s − 3.03·25-s − 5.65·27-s − 4.38·29-s + 3.62·31-s − 6.39·33-s + 1.40·35-s + 2.93·37-s + 0.729·41-s + 11.7·43-s − 1.25·45-s + 3.07·47-s + 49-s + 0.210·51-s + 4.22·53-s − 6.18·55-s + 2.89·57-s + 12.7·59-s + ⋯ |
| L(s) = 1 | + 0.837·3-s + 0.627·5-s + 0.377·7-s − 0.298·9-s − 1.32·11-s + 0.525·15-s + 0.0351·17-s + 0.457·19-s + 0.316·21-s + 1.26·23-s − 0.606·25-s − 1.08·27-s − 0.814·29-s + 0.650·31-s − 1.11·33-s + 0.237·35-s + 0.483·37-s + 0.113·41-s + 1.79·43-s − 0.187·45-s + 0.448·47-s + 0.142·49-s + 0.0294·51-s + 0.580·53-s − 0.833·55-s + 0.383·57-s + 1.65·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.970887594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.970887594\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.45T + 3T^{2} \) |
| 5 | \( 1 - 1.40T + 5T^{2} \) |
| 11 | \( 1 + 4.40T + 11T^{2} \) |
| 17 | \( 1 - 0.145T + 17T^{2} \) |
| 19 | \( 1 - 1.99T + 19T^{2} \) |
| 23 | \( 1 - 6.06T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 3.62T + 31T^{2} \) |
| 37 | \( 1 - 2.93T + 37T^{2} \) |
| 41 | \( 1 - 0.729T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 3.07T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 2.86T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 8.81T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 6.12T + 97T^{2} \) |
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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
−7.67932565553885172737233911263, −7.34798676090410695377103804640, −6.21701584669312781918968084500, −5.56229324504932791428703589266, −5.09195854875254574437160307120, −4.13923282897093292165640406362, −3.22325777467382068542382810176, −2.55554748581762357858421784661, −2.03714393148241188051042324420, −0.77299914601925933221362458295,
0.77299914601925933221362458295, 2.03714393148241188051042324420, 2.55554748581762357858421784661, 3.22325777467382068542382810176, 4.13923282897093292165640406362, 5.09195854875254574437160307120, 5.56229324504932791428703589266, 6.21701584669312781918968084500, 7.34798676090410695377103804640, 7.67932565553885172737233911263
