| L(s) = 1 | + 3-s + 3.37·5-s + 3.37·7-s + 9-s + 11-s − 13-s + 3.37·15-s − 4·17-s + 6.74·19-s + 3.37·21-s + 3.37·23-s + 6.37·25-s + 27-s − 3.37·29-s − 8.74·31-s + 33-s + 11.3·35-s − 2·37-s − 39-s + 5.37·41-s + 2.62·43-s + 3.37·45-s + 4·47-s + 4.37·49-s − 4·51-s − 10·53-s + 3.37·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.50·5-s + 1.27·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s + 0.870·15-s − 0.970·17-s + 1.54·19-s + 0.735·21-s + 0.703·23-s + 1.27·25-s + 0.192·27-s − 0.626·29-s − 1.57·31-s + 0.174·33-s + 1.92·35-s − 0.328·37-s − 0.160·39-s + 0.839·41-s + 0.400·43-s + 0.502·45-s + 0.583·47-s + 0.624·49-s − 0.560·51-s − 1.37·53-s + 0.454·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.388804658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.388804658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 - 3.37T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 - 2.62T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 8.11T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.925420226412624068642243432977, −7.31138508710785040308044885915, −6.66154573941343039170716073729, −5.62494640243060875777466899097, −5.26481254065914092328052368834, −4.48660913779082125170326539439, −3.50057662720215514638876296515, −2.46786281578398541566464297111, −1.89198308038529015901832397676, −1.15293157346452956141989539434,
1.15293157346452956141989539434, 1.89198308038529015901832397676, 2.46786281578398541566464297111, 3.50057662720215514638876296515, 4.48660913779082125170326539439, 5.26481254065914092328052368834, 5.62494640243060875777466899097, 6.66154573941343039170716073729, 7.31138508710785040308044885915, 7.925420226412624068642243432977
