L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 11-s + 6·13-s + 16-s + 2·17-s − 4·19-s + 2·20-s − 22-s − 4·23-s − 25-s − 6·26-s − 6·29-s − 32-s − 2·34-s + 6·37-s + 4·38-s − 2·40-s − 6·41-s + 4·43-s + 44-s + 4·46-s − 12·47-s + 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 0.301·11-s + 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s − 1.17·26-s − 1.11·29-s − 0.176·32-s − 0.342·34-s + 0.986·37-s + 0.648·38-s − 0.316·40-s − 0.937·41-s + 0.609·43-s + 0.150·44-s + 0.589·46-s − 1.75·47-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.924145028\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.924145028\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.932935680349574946698629127528, −6.89602460657242384923499783608, −6.28924205494827979239593212209, −5.90660471214004423532671338501, −5.14182090257164174954778440071, −3.94579611699495679822947403941, −3.48324158491306848074738148539, −2.22937020986074254273749556155, −1.74167359579764026972468343842, −0.75158390180470276062868800487,
0.75158390180470276062868800487, 1.74167359579764026972468343842, 2.22937020986074254273749556155, 3.48324158491306848074738148539, 3.94579611699495679822947403941, 5.14182090257164174954778440071, 5.90660471214004423532671338501, 6.28924205494827979239593212209, 6.89602460657242384923499783608, 7.932935680349574946698629127528