| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s + 6·11-s + 13-s + 15-s − 17-s + 2·19-s + 2·21-s + 4·23-s + 25-s + 27-s − 2·29-s − 6·31-s + 6·33-s + 2·35-s + 2·37-s + 39-s + 2·41-s + 8·43-s + 45-s + 10·47-s − 3·49-s − 51-s + 10·53-s + 6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.458·19-s + 0.436·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.07·31-s + 1.04·33-s + 0.338·35-s + 0.328·37-s + 0.160·39-s + 0.312·41-s + 1.21·43-s + 0.149·45-s + 1.45·47-s − 3/7·49-s − 0.140·51-s + 1.37·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.054506077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.054506077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.04653410360827, −12.61581731696040, −11.99891844308732, −11.62532113525045, −11.15136756553250, −10.72222949987554, −10.20100192662697, −9.493425820516479, −9.094850817198253, −8.975921528187467, −8.437354954191139, −7.630257298454648, −7.395555244592393, −6.836189704155405, −6.270387734461596, −5.813150193992509, −5.204920776999470, −4.647369051211940, −4.032469598759100, −3.704224197592908, −3.087318507544164, −2.221839548315032, −1.947050553249707, −1.109089082825222, −0.8359094193013993,
0.8359094193013993, 1.109089082825222, 1.947050553249707, 2.221839548315032, 3.087318507544164, 3.704224197592908, 4.032469598759100, 4.647369051211940, 5.204920776999470, 5.813150193992509, 6.270387734461596, 6.836189704155405, 7.395555244592393, 7.630257298454648, 8.437354954191139, 8.975921528187467, 9.094850817198253, 9.493425820516479, 10.20100192662697, 10.72222949987554, 11.15136756553250, 11.62532113525045, 11.99891844308732, 12.61581731696040, 13.04653410360827
