| L(s) = 1 | − 3-s + 1.74·5-s + 4.53·7-s + 9-s − 11-s + 13-s − 1.74·15-s − 1.63·17-s − 0.534·19-s − 4.53·21-s + 7.60·23-s − 1.95·25-s − 27-s − 1.21·29-s + 2.67·31-s + 33-s + 7.91·35-s − 9.91·37-s − 39-s + 3.37·41-s − 3.63·43-s + 1.74·45-s + 9.48·47-s + 13.5·49-s + 1.63·51-s − 3.57·53-s − 1.74·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.780·5-s + 1.71·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s − 0.450·15-s − 0.395·17-s − 0.122·19-s − 0.989·21-s + 1.58·23-s − 0.391·25-s − 0.192·27-s − 0.224·29-s + 0.480·31-s + 0.174·33-s + 1.33·35-s − 1.62·37-s − 0.160·39-s + 0.527·41-s − 0.553·43-s + 0.260·45-s + 1.38·47-s + 1.93·49-s + 0.228·51-s − 0.491·53-s − 0.235·55-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
| \(L(1)\) |
\(\approx\) |
\(2.550455187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.550455187\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.74T + 5T^{2} \) |
| 7 | \( 1 - 4.53T + 7T^{2} \) |
| 17 | \( 1 + 1.63T + 17T^{2} \) |
| 19 | \( 1 + 0.534T + 19T^{2} \) |
| 23 | \( 1 - 7.60T + 23T^{2} \) |
| 29 | \( 1 + 1.21T + 29T^{2} \) |
| 31 | \( 1 - 2.67T + 31T^{2} \) |
| 37 | \( 1 + 9.91T + 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 + 3.63T + 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 + 3.57T + 53T^{2} \) |
| 59 | \( 1 - 2.64T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 3.06T + 71T^{2} \) |
| 73 | \( 1 - 4.95T + 73T^{2} \) |
| 79 | \( 1 - 1.72T + 79T^{2} \) |
| 83 | \( 1 + 1.35T + 83T^{2} \) |
| 89 | \( 1 - 6.39T + 89T^{2} \) |
| 97 | \( 1 + 2.93T + 97T^{2} \) |
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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
−7.978026712640444263871169994589, −7.19791480038701363399291175378, −6.58454085803582791423679186860, −5.61567897880384939575105516275, −5.21598595998499094436677325749, −4.63534042877687707171738311504, −3.72507466828502982881287949825, −2.44483609880182626691745167249, −1.76379012993947609705281228783, −0.891932050236332853054426865322,
0.891932050236332853054426865322, 1.76379012993947609705281228783, 2.44483609880182626691745167249, 3.72507466828502982881287949825, 4.63534042877687707171738311504, 5.21598595998499094436677325749, 5.61567897880384939575105516275, 6.58454085803582791423679186860, 7.19791480038701363399291175378, 7.978026712640444263871169994589
