| L(s) = 1 | + 3-s + 0.279·5-s + 4.36·7-s + 9-s − 11-s − 13-s + 0.279·15-s + 2.64·17-s − 4.58·19-s + 4.36·21-s − 1.80·23-s − 4.92·25-s + 27-s + 3.30·29-s + 1.50·31-s − 33-s + 1.22·35-s − 0.559·37-s − 39-s + 6.36·41-s + 6.97·43-s + 0.279·45-s − 2.73·47-s + 12.0·49-s + 2.64·51-s + 2.99·53-s − 0.279·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.125·5-s + 1.65·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s + 0.0722·15-s + 0.641·17-s − 1.05·19-s + 0.952·21-s − 0.376·23-s − 0.984·25-s + 0.192·27-s + 0.614·29-s + 0.269·31-s − 0.174·33-s + 0.206·35-s − 0.0920·37-s − 0.160·39-s + 0.994·41-s + 1.06·43-s + 0.0417·45-s − 0.398·47-s + 1.72·49-s + 0.370·51-s + 0.411·53-s − 0.0377·55-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
| \(L(1)\) |
\(\approx\) |
\(3.236680786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.236680786\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.279T + 5T^{2} \) |
| 7 | \( 1 - 4.36T + 7T^{2} \) |
| 17 | \( 1 - 2.64T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 23 | \( 1 + 1.80T + 23T^{2} \) |
| 29 | \( 1 - 3.30T + 29T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 37 | \( 1 + 0.559T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 - 6.97T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 - 2.99T + 53T^{2} \) |
| 59 | \( 1 - 8.39T + 59T^{2} \) |
| 61 | \( 1 + 1.10T + 61T^{2} \) |
| 67 | \( 1 - 4.39T + 67T^{2} \) |
| 71 | \( 1 - 2.99T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 2.49T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−8.062503805551354975770420010132, −7.54500046320063816238303909798, −6.64018581090668569659762554102, −5.74497649761612070621231836771, −5.05565054609485829651572473322, −4.36176015423648488926371019166, −3.70104113418123551034458910891, −2.44074017873941600848111925102, −2.01779302182352517373450885736, −0.923436639841302740364866937713,
0.923436639841302740364866937713, 2.01779302182352517373450885736, 2.44074017873941600848111925102, 3.70104113418123551034458910891, 4.36176015423648488926371019166, 5.05565054609485829651572473322, 5.74497649761612070621231836771, 6.64018581090668569659762554102, 7.54500046320063816238303909798, 8.062503805551354975770420010132
