| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 1.61·11-s + 6.31·13-s − 15-s + 7.92·17-s + 2.38·19-s − 21-s + 6.70·23-s + 25-s + 27-s − 7.92·29-s − 9.08·31-s + 1.61·33-s + 35-s + 7.92·37-s + 6.31·39-s − 1.22·41-s − 5.92·43-s − 45-s − 6.70·47-s + 49-s + 7.92·51-s + 3.61·53-s − 1.61·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s + 0.486·11-s + 1.75·13-s − 0.258·15-s + 1.92·17-s + 0.547·19-s − 0.218·21-s + 1.39·23-s + 0.200·25-s + 0.192·27-s − 1.47·29-s − 1.63·31-s + 0.280·33-s + 0.169·35-s + 1.30·37-s + 1.01·39-s − 0.191·41-s − 0.903·43-s − 0.149·45-s − 0.977·47-s + 0.142·49-s + 1.10·51-s + 0.496·53-s − 0.217·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.838754230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.838754230\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 - 6.31T + 13T^{2} \) |
| 17 | \( 1 - 7.92T + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 + 9.08T + 31T^{2} \) |
| 37 | \( 1 - 7.92T + 37T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 + 5.92T + 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 - 3.61T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 - 9.14T + 67T^{2} \) |
| 71 | \( 1 + 8.31T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 0.775T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−7.998903756370211764289446653645, −7.35384544207410665935817652452, −6.72702806781821032083231068389, −5.78955157306591278285901809922, −5.28563772281094274399418963648, −4.05171279846493244156461497840, −3.48301957242344771133213188762, −3.11400355674867263286728782482, −1.66036574923469271938003467680, −0.918808188725997685440904254958,
0.918808188725997685440904254958, 1.66036574923469271938003467680, 3.11400355674867263286728782482, 3.48301957242344771133213188762, 4.05171279846493244156461497840, 5.28563772281094274399418963648, 5.78955157306591278285901809922, 6.72702806781821032083231068389, 7.35384544207410665935817652452, 7.998903756370211764289446653645
