L(s) = 1 | + 2.56·3-s − 5-s − 4.56·7-s + 3.56·9-s + 4·11-s + 2.56·13-s − 2.56·15-s − 0.561·17-s + 19-s − 11.6·21-s + 5.68·23-s + 25-s + 1.43·27-s − 3.43·29-s + 5.12·31-s + 10.2·33-s + 4.56·35-s − 1.12·37-s + 6.56·39-s + 8.24·41-s + 2·43-s − 3.56·45-s − 3.12·47-s + 13.8·49-s − 1.43·51-s + 6.56·53-s − 4·55-s + ⋯ |
L(s) = 1 | + 1.47·3-s − 0.447·5-s − 1.72·7-s + 1.18·9-s + 1.20·11-s + 0.710·13-s − 0.661·15-s − 0.136·17-s + 0.229·19-s − 2.54·21-s + 1.18·23-s + 0.200·25-s + 0.276·27-s − 0.638·29-s + 0.920·31-s + 1.78·33-s + 0.771·35-s − 0.184·37-s + 1.05·39-s + 1.28·41-s + 0.304·43-s − 0.530·45-s − 0.455·47-s + 1.97·49-s − 0.201·51-s + 0.901·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.678057748\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.678057748\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 + 4.56T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2.56T + 13T^{2} \) |
| 17 | \( 1 + 0.561T + 17T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 + 3.43T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 - 6.56T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 4.87T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−8.895318358094823467828782393939, −8.094466549504147361129515408084, −7.22262196112032373566630806190, −6.63916936997000433512436559977, −5.91786885671210340845076591957, −4.46522482451246616473928113876, −3.54985649169236479077483425639, −3.34793456630333448937126847499, −2.35582090438890933964225614140, −0.948917446234411268788742012051,
0.948917446234411268788742012051, 2.35582090438890933964225614140, 3.34793456630333448937126847499, 3.54985649169236479077483425639, 4.46522482451246616473928113876, 5.91786885671210340845076591957, 6.63916936997000433512436559977, 7.22262196112032373566630806190, 8.094466549504147361129515408084, 8.895318358094823467828782393939