| L(s) = 1 | − 0.904·2-s − 1.18·4-s + 0.384·5-s + 3.36·7-s + 2.87·8-s − 0.348·10-s + 0.382·11-s + 0.191·13-s − 3.04·14-s − 0.240·16-s − 3.70·17-s + 4.49·19-s − 0.454·20-s − 0.346·22-s + 4.36·23-s − 4.85·25-s − 0.172·26-s − 3.97·28-s − 2.79·29-s − 5.53·32-s + 3.35·34-s + 1.29·35-s + 5.21·37-s − 4.06·38-s + 1.10·40-s − 12.7·41-s − 0.665·43-s + ⋯ |
| L(s) = 1 | − 0.639·2-s − 0.590·4-s + 0.172·5-s + 1.27·7-s + 1.01·8-s − 0.110·10-s + 0.115·11-s + 0.0530·13-s − 0.814·14-s − 0.0601·16-s − 0.898·17-s + 1.03·19-s − 0.101·20-s − 0.0737·22-s + 0.910·23-s − 0.970·25-s − 0.0339·26-s − 0.752·28-s − 0.519·29-s − 0.979·32-s + 0.574·34-s + 0.219·35-s + 0.857·37-s − 0.659·38-s + 0.175·40-s − 1.98·41-s − 0.101·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.502175687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.502175687\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 0.904T + 2T^{2} \) |
| 5 | \( 1 - 0.384T + 5T^{2} \) |
| 7 | \( 1 - 3.36T + 7T^{2} \) |
| 11 | \( 1 - 0.382T + 11T^{2} \) |
| 13 | \( 1 - 0.191T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 - 4.49T + 19T^{2} \) |
| 23 | \( 1 - 4.36T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 37 | \( 1 - 5.21T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 + 0.665T + 43T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 5.30T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 5.03T + 79T^{2} \) |
| 83 | \( 1 - 9.95T + 83T^{2} \) |
| 89 | \( 1 - 9.90T + 89T^{2} \) |
| 97 | \( 1 - 4.66T + 97T^{2} \) |
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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
−7.891931770203147082940886688751, −7.30772438023093609944076887734, −6.58660461770387115309660254409, −5.29754511294839444685539283106, −5.23274140839561273857784808591, −4.26671502094792176625383009063, −3.65087122397656127352738465647, −2.35139908114804835309380934780, −1.57537747160737814264605145868, −0.70893879202268204051390560637,
0.70893879202268204051390560637, 1.57537747160737814264605145868, 2.35139908114804835309380934780, 3.65087122397656127352738465647, 4.26671502094792176625383009063, 5.23274140839561273857784808591, 5.29754511294839444685539283106, 6.58660461770387115309660254409, 7.30772438023093609944076887734, 7.891931770203147082940886688751
