| L(s) = 1 | + 5.47·3-s + 14.8·5-s − 32.4·7-s + 2.94·9-s + 71.1·13-s + 81.4·15-s + 71.0·17-s − 134.·19-s − 177.·21-s − 96.6·23-s + 96.6·25-s − 131.·27-s − 177.·29-s − 71.6·31-s − 483.·35-s − 161.·37-s + 389.·39-s − 184.·41-s − 60.7·43-s + 43.8·45-s + 180.·47-s + 711.·49-s + 388.·51-s − 592.·53-s − 734.·57-s + 180.·59-s + 433.·61-s + ⋯ |
| L(s) = 1 | + 1.05·3-s + 1.33·5-s − 1.75·7-s + 0.109·9-s + 1.51·13-s + 1.40·15-s + 1.01·17-s − 1.62·19-s − 1.84·21-s − 0.876·23-s + 0.773·25-s − 0.938·27-s − 1.13·29-s − 0.415·31-s − 2.33·35-s − 0.717·37-s + 1.59·39-s − 0.701·41-s − 0.215·43-s + 0.145·45-s + 0.560·47-s + 2.07·49-s + 1.06·51-s − 1.53·53-s − 1.70·57-s + 0.398·59-s + 0.908·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 5.47T + 27T^{2} \) |
| 5 | \( 1 - 14.8T + 125T^{2} \) |
| 7 | \( 1 + 32.4T + 343T^{2} \) |
| 13 | \( 1 - 71.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 134.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 177.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 71.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 184.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 60.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 180.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 592.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 180.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 433.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 149.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 754.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 419.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 744.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 692.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 312.T + 9.12e5T^{2} \) |
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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
−8.785992689605809540426527097655, −7.80738603230664396553114220709, −6.64103983402027649915696852858, −6.10202659077132739686268606409, −5.57402363760011354603419166132, −3.83822218244963508810533856023, −3.40611067421553594683222472925, −2.42279116202928999228091227971, −1.61104856366821031307406691851, 0,
1.61104856366821031307406691851, 2.42279116202928999228091227971, 3.40611067421553594683222472925, 3.83822218244963508810533856023, 5.57402363760011354603419166132, 6.10202659077132739686268606409, 6.64103983402027649915696852858, 7.80738603230664396553114220709, 8.785992689605809540426527097655
