| L(s) = 1 | + 3-s − 2·4-s − 2·9-s − 11-s − 2·12-s − 4·13-s + 4·16-s − 6·17-s − 2·19-s − 3·23-s − 5·27-s − 6·29-s − 5·31-s − 33-s + 4·36-s − 11·37-s − 4·39-s − 6·41-s − 8·43-s + 2·44-s + 4·48-s − 6·51-s + 8·52-s + 6·53-s − 2·57-s + 9·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 2/3·9-s − 0.301·11-s − 0.577·12-s − 1.10·13-s + 16-s − 1.45·17-s − 0.458·19-s − 0.625·23-s − 0.962·27-s − 1.11·29-s − 0.898·31-s − 0.174·33-s + 2/3·36-s − 1.80·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s + 0.301·44-s + 0.577·48-s − 0.840·51-s + 1.10·52-s + 0.824·53-s − 0.264·57-s + 1.17·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2624895205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2624895205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−16.29660548477604, −15.35084509341807, −15.03626342130959, −14.48931159741421, −13.98811517575641, −13.36269542865518, −13.07108495824734, −12.34799792776457, −11.71613291794205, −11.08034362512046, −10.28725750471382, −9.843778705801598, −9.135605107620202, −8.629746091664673, −8.338159078086142, −7.461933279963957, −6.925805892512313, −5.995268524264120, −5.204151935226225, −4.888359947633878, −3.853168666485327, −3.502358000052814, −2.382405082871667, −1.907555374550167, −0.2069772519885212,
0.2069772519885212, 1.907555374550167, 2.382405082871667, 3.502358000052814, 3.853168666485327, 4.888359947633878, 5.204151935226225, 5.995268524264120, 6.925805892512313, 7.461933279963957, 8.338159078086142, 8.629746091664673, 9.135605107620202, 9.843778705801598, 10.28725750471382, 11.08034362512046, 11.71613291794205, 12.34799792776457, 13.07108495824734, 13.36269542865518, 13.98811517575641, 14.48931159741421, 15.03626342130959, 15.35084509341807, 16.29660548477604
