| L(s) = 1 | + 3-s + 0.449·5-s + 2·7-s + 9-s − 11-s + 13-s + 0.449·15-s − 6.44·17-s − 6.89·19-s + 2·21-s + 2.89·23-s − 4.79·25-s + 27-s + 1.55·29-s − 6.44·31-s − 33-s + 0.898·35-s + 2·37-s + 39-s + 0.898·41-s + 0.449·43-s + 0.449·45-s + 4.89·47-s − 3·49-s − 6.44·51-s − 6·53-s − 0.449·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.201·5-s + 0.755·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s + 0.116·15-s − 1.56·17-s − 1.58·19-s + 0.436·21-s + 0.604·23-s − 0.959·25-s + 0.192·27-s + 0.287·29-s − 1.15·31-s − 0.174·33-s + 0.151·35-s + 0.328·37-s + 0.160·39-s + 0.140·41-s + 0.0685·43-s + 0.0670·45-s + 0.714·47-s − 0.428·49-s − 0.903·51-s − 0.824·53-s − 0.0606·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.449T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 + 6.89T + 19T^{2} \) |
| 23 | \( 1 - 2.89T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 0.898T + 41T^{2} \) |
| 43 | \( 1 - 0.449T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 1.55T + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 + 0.898T + 73T^{2} \) |
| 79 | \( 1 - 9.34T + 79T^{2} \) |
| 83 | \( 1 + 9.79T + 83T^{2} \) |
| 89 | \( 1 + 4.44T + 89T^{2} \) |
| 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
−7.76319348219339926835955019867, −6.93910561814564345404143395612, −6.28982308354127300045166465114, −5.49236546260907539230826188780, −4.47974541509801766756589621559, −4.21559310691597746324853040081, −3.05223049983477718929779064089, −2.17892950594738816622263988567, −1.61113803699920965864417313646, 0,
1.61113803699920965864417313646, 2.17892950594738816622263988567, 3.05223049983477718929779064089, 4.21559310691597746324853040081, 4.47974541509801766756589621559, 5.49236546260907539230826188780, 6.28982308354127300045166465114, 6.93910561814564345404143395612, 7.76319348219339926835955019867
