| L(s) = 1 | − 1.61·2-s + 0.618·4-s + 0.874·5-s + 2.23·8-s − 1.41·10-s − 11-s − 1.74·13-s − 4.85·16-s − 3.16·17-s + 5.45·19-s + 0.540·20-s + 1.61·22-s + 2.76·23-s − 4.23·25-s + 2.82·26-s + 3.23·29-s + 0.333·31-s + 3.38·32-s + 5.11·34-s + 2.70·37-s − 8.81·38-s + 1.95·40-s − 10.0·41-s − 11.9·43-s − 0.618·44-s − 4.47·46-s + 12.5·47-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s + 0.390·5-s + 0.790·8-s − 0.447·10-s − 0.301·11-s − 0.484·13-s − 1.21·16-s − 0.766·17-s + 1.25·19-s + 0.120·20-s + 0.344·22-s + 0.576·23-s − 0.847·25-s + 0.554·26-s + 0.600·29-s + 0.0599·31-s + 0.597·32-s + 0.877·34-s + 0.445·37-s − 1.43·38-s + 0.309·40-s − 1.56·41-s − 1.82·43-s − 0.0931·44-s − 0.659·46-s + 1.82·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 - 0.874T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 5.45T + 19T^{2} \) |
| 23 | \( 1 - 2.76T + 23T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 - 0.333T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 - 8.15T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 7.47T + 67T^{2} \) |
| 71 | \( 1 - 3.47T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 7.19T + 83T^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 - 9.35T + 97T^{2} \) |
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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
−8.265704099124665194952861350369, −7.39616860089096840741800477087, −7.00023426831918588784974880024, −5.94309058596093700632412785862, −5.07805143340507795157227559589, −4.41322336534226989524796469781, −3.19983593181019503833208028765, −2.19108860104994771447599574560, −1.24791344974786545368513351054, 0,
1.24791344974786545368513351054, 2.19108860104994771447599574560, 3.19983593181019503833208028765, 4.41322336534226989524796469781, 5.07805143340507795157227559589, 5.94309058596093700632412785862, 7.00023426831918588784974880024, 7.39616860089096840741800477087, 8.265704099124665194952861350369
