| L(s) = 1 | − 5-s − 6·11-s − 2·13-s − 6·17-s − 8·19-s − 3·23-s + 25-s − 3·29-s − 2·31-s + 8·37-s − 3·41-s + 5·43-s − 12·53-s + 6·55-s + 61-s + 2·65-s − 7·67-s + 10·73-s − 4·79-s + 3·83-s + 6·85-s − 3·89-s + 8·95-s + 10·97-s − 3·101-s + 7·103-s + 3·107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.80·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 0.625·23-s + 1/5·25-s − 0.557·29-s − 0.359·31-s + 1.31·37-s − 0.468·41-s + 0.762·43-s − 1.64·53-s + 0.809·55-s + 0.128·61-s + 0.248·65-s − 0.855·67-s + 1.17·73-s − 0.450·79-s + 0.329·83-s + 0.650·85-s − 0.317·89-s + 0.820·95-s + 1.01·97-s − 0.298·101-s + 0.689·103-s + 0.290·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4181548244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4181548244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.77468326399584550521827841367, −7.20111926956575288966148646324, −6.34142829172486695403193989702, −5.77091786358778297474480067197, −4.67021280813719951130691094523, −4.53586003248224039106093854886, −3.45679418594533079489720780262, −2.43909193961244322183180369905, −2.07696906998804114155894339696, −0.28745544287901866526726671660,
0.28745544287901866526726671660, 2.07696906998804114155894339696, 2.43909193961244322183180369905, 3.45679418594533079489720780262, 4.53586003248224039106093854886, 4.67021280813719951130691094523, 5.77091786358778297474480067197, 6.34142829172486695403193989702, 7.20111926956575288966148646324, 7.77468326399584550521827841367
