| L(s) = 1 | − 1.56·3-s + 2.56·5-s − 3·7-s − 0.561·9-s + 0.561·11-s + 1.56·13-s − 4·15-s + 0.123·17-s + 19-s + 4.68·21-s − 5.56·23-s + 1.56·25-s + 5.56·27-s − 4.68·29-s − 9.12·31-s − 0.876·33-s − 7.68·35-s − 7.12·37-s − 2.43·39-s + 4·41-s + 5.43·43-s − 1.43·45-s + 3.68·47-s + 2·49-s − 0.192·51-s + 4.43·53-s + 1.43·55-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 1.14·5-s − 1.13·7-s − 0.187·9-s + 0.169·11-s + 0.433·13-s − 1.03·15-s + 0.0298·17-s + 0.229·19-s + 1.02·21-s − 1.15·23-s + 0.312·25-s + 1.07·27-s − 0.869·29-s − 1.63·31-s − 0.152·33-s − 1.29·35-s − 1.17·37-s − 0.390·39-s + 0.624·41-s + 0.829·43-s − 0.214·45-s + 0.537·47-s + 0.285·49-s − 0.0269·51-s + 0.609·53-s + 0.193·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 0.561T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 0.123T + 17T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 + 4.43T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 + 0.438T + 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + 9.24T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.401534569465140448478223180627, −8.820604068959892843241852593888, −7.45554932226532611543732923011, −6.55352104764641269045717217135, −5.74448315765810847195685347969, −5.62603650770663543413790010552, −4.10152857220081832419368412288, −2.98956207569121307769104942490, −1.70831221281079586954224147811, 0,
1.70831221281079586954224147811, 2.98956207569121307769104942490, 4.10152857220081832419368412288, 5.62603650770663543413790010552, 5.74448315765810847195685347969, 6.55352104764641269045717217135, 7.45554932226532611543732923011, 8.820604068959892843241852593888, 9.401534569465140448478223180627
