| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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
−7.40568310000396117205430158943, −6.82670665512494985685724823175, −6.05599848539531165464919253010, −5.33534673161326905561454654631, −4.23993823902556171283183903115, −3.78005753551973352786569372319, −3.15109865225258183342637939816, −2.29776944352015095140617108493, −1.30722785104407642348402453013, 0,
1.30722785104407642348402453013, 2.29776944352015095140617108493, 3.15109865225258183342637939816, 3.78005753551973352786569372319, 4.23993823902556171283183903115, 5.33534673161326905561454654631, 6.05599848539531165464919253010, 6.82670665512494985685724823175, 7.40568310000396117205430158943
