| L(s) = 1 | − 2·5-s + 7-s + 2·13-s + 6·17-s + 4·19-s + 4·23-s − 25-s + 6·29-s − 8·31-s − 2·35-s − 10·37-s − 10·41-s − 12·43-s + 8·47-s + 49-s − 6·53-s − 4·59-s + 10·61-s − 4·65-s + 12·67-s − 4·71-s − 2·73-s − 8·79-s + 4·83-s − 12·85-s − 6·89-s + 2·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.338·35-s − 1.64·37-s − 1.56·41-s − 1.82·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.520·59-s + 1.28·61-s − 0.496·65-s + 1.46·67-s − 0.474·71-s − 0.234·73-s − 0.900·79-s + 0.439·83-s − 1.30·85-s − 0.635·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.021058812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.021058812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.27579301326038, −13.91333182010555, −13.29308791936569, −12.71940129226222, −12.08783422534199, −11.84277230812225, −11.34859427913215, −10.81186348821723, −10.14146882920515, −9.873416739026732, −8.990111205728011, −8.544538770417404, −8.122928318604748, −7.494414572950458, −7.116734764271468, −6.541415491026936, −5.660134793382824, −5.223917608283388, −4.789148705102867, −3.863725076968595, −3.400667589205375, −3.103901829130148, −1.908373581571368, −1.320461899774931, −0.5169443104941732,
0.5169443104941732, 1.320461899774931, 1.908373581571368, 3.103901829130148, 3.400667589205375, 3.863725076968595, 4.789148705102867, 5.223917608283388, 5.660134793382824, 6.541415491026936, 7.116734764271468, 7.494414572950458, 8.122928318604748, 8.544538770417404, 8.990111205728011, 9.873416739026732, 10.14146882920515, 10.81186348821723, 11.34859427913215, 11.84277230812225, 12.08783422534199, 12.71940129226222, 13.29308791936569, 13.91333182010555, 14.27579301326038
