| L(s) = 1 | + 3.27·3-s + 6.27·5-s + 18.0·7-s − 16.2·9-s − 33.9·11-s + 3.07·13-s + 20.5·15-s − 14.2·17-s − 19·19-s + 59.2·21-s − 114.·23-s − 85.6·25-s − 141.·27-s + 34.5·29-s − 107.·31-s − 111.·33-s + 113.·35-s + 181.·37-s + 10.0·39-s − 444.·41-s + 120.·43-s − 102.·45-s − 306.·47-s − 15.4·49-s − 46.6·51-s − 115.·53-s − 212.·55-s + ⋯ |
| L(s) = 1 | + 0.630·3-s + 0.561·5-s + 0.977·7-s − 0.602·9-s − 0.929·11-s + 0.0656·13-s + 0.353·15-s − 0.203·17-s − 0.229·19-s + 0.615·21-s − 1.03·23-s − 0.685·25-s − 1.01·27-s + 0.221·29-s − 0.623·31-s − 0.586·33-s + 0.548·35-s + 0.807·37-s + 0.0413·39-s − 1.69·41-s + 0.426·43-s − 0.338·45-s − 0.950·47-s − 0.0449·49-s − 0.128·51-s − 0.300·53-s − 0.521·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 3.27T + 27T^{2} \) |
| 5 | \( 1 - 6.27T + 125T^{2} \) |
| 7 | \( 1 - 18.0T + 343T^{2} \) |
| 11 | \( 1 + 33.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.07T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 34.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 107.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 120.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 115.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 161.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 81.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 773.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 148.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 557.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 768.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 457.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.50e3T + 9.12e5T^{2} \) |
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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
−8.837288451997550384778094492909, −8.103161055469253879596759956899, −7.66244134883072423161989835727, −6.33353283813303207043393214044, −5.52723084244230813672143173887, −4.72067022483333522048769755595, −3.53781171579627802275757303231, −2.42519103754010673657380446192, −1.73539415815282408875612601094, 0,
1.73539415815282408875612601094, 2.42519103754010673657380446192, 3.53781171579627802275757303231, 4.72067022483333522048769755595, 5.52723084244230813672143173887, 6.33353283813303207043393214044, 7.66244134883072423161989835727, 8.103161055469253879596759956899, 8.837288451997550384778094492909
