| L(s) = 1 | − 3·3-s + 5·5-s + 3.34·7-s + 9·9-s − 38.6·11-s − 18·13-s − 15·15-s + 106.·17-s + 5.31·19-s − 10.0·21-s + 23·23-s + 25·25-s − 27·27-s − 241.·29-s − 70.1·31-s + 116.·33-s + 16.7·35-s − 49.2·37-s + 54·39-s − 68.0·41-s + 476.·43-s + 45·45-s + 496.·47-s − 331.·49-s − 320.·51-s − 504.·53-s − 193.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.180·7-s + 0.333·9-s − 1.06·11-s − 0.384·13-s − 0.258·15-s + 1.52·17-s + 0.0641·19-s − 0.104·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 1.54·29-s − 0.406·31-s + 0.612·33-s + 0.0807·35-s − 0.218·37-s + 0.221·39-s − 0.259·41-s + 1.68·43-s + 0.149·45-s + 1.54·47-s − 0.967·49-s − 0.878·51-s − 1.30·53-s − 0.474·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 - 3.34T + 343T^{2} \) |
| 11 | \( 1 + 38.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18T + 2.19e3T^{2} \) |
| 17 | \( 1 - 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.31T + 6.85e3T^{2} \) |
| 29 | \( 1 + 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 70.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 49.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 68.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 476.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 496.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 504.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 360.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 433.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 524.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 152.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 902.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 261.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.78e3T + 9.12e5T^{2} \) |
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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
−8.939547825010979642247186471591, −7.66285301665747731347125801825, −7.43871557642529243815749619756, −6.07981007015561148439864452300, −5.50863430415299920133562854033, −4.81425614266033192083782990637, −3.56968279244142469207351638125, −2.46239734227024303880779208214, −1.29585117441927010468347586065, 0,
1.29585117441927010468347586065, 2.46239734227024303880779208214, 3.56968279244142469207351638125, 4.81425614266033192083782990637, 5.50863430415299920133562854033, 6.07981007015561148439864452300, 7.43871557642529243815749619756, 7.66285301665747731347125801825, 8.939547825010979642247186471591
