| L(s) = 1 | + 2-s + 3-s + 4-s − 3.18·5-s + 6-s + 8-s + 9-s − 3.18·10-s − 1.67·11-s + 12-s + 2.27·13-s − 3.18·15-s + 16-s + 0.367·17-s + 18-s − 3.41·19-s − 3.18·20-s − 1.67·22-s − 23-s + 24-s + 5.13·25-s + 2.27·26-s + 27-s + 3.44·29-s − 3.18·30-s − 3.41·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.42·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.00·10-s − 0.504·11-s + 0.288·12-s + 0.630·13-s − 0.822·15-s + 0.250·16-s + 0.0892·17-s + 0.235·18-s − 0.783·19-s − 0.711·20-s − 0.356·22-s − 0.208·23-s + 0.204·24-s + 1.02·25-s + 0.445·26-s + 0.192·27-s + 0.639·29-s − 0.581·30-s − 0.613·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3.18T + 5T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 - 2.27T + 13T^{2} \) |
| 17 | \( 1 - 0.367T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 29 | \( 1 - 3.44T + 29T^{2} \) |
| 31 | \( 1 + 3.41T + 31T^{2} \) |
| 37 | \( 1 + 0.0953T + 37T^{2} \) |
| 41 | \( 1 + 0.681T + 41T^{2} \) |
| 43 | \( 1 + 7.94T + 43T^{2} \) |
| 47 | \( 1 + 1.64T + 47T^{2} \) |
| 53 | \( 1 - 9.69T + 53T^{2} \) |
| 59 | \( 1 + 3.15T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 5.19T + 67T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 4.32T + 79T^{2} \) |
| 83 | \( 1 - 4.74T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
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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
−7.61344882019207341088999596815, −7.01757100537428134670466569227, −6.27904020801473375523990347880, −5.35777364586143871321441040931, −4.50698523118668515056602931741, −3.97752352004984294314337263818, −3.33034933088005993313489740902, −2.59841117830723502926013070224, −1.46022375418883043961928680402, 0,
1.46022375418883043961928680402, 2.59841117830723502926013070224, 3.33034933088005993313489740902, 3.97752352004984294314337263818, 4.50698523118668515056602931741, 5.35777364586143871321441040931, 6.27904020801473375523990347880, 7.01757100537428134670466569227, 7.61344882019207341088999596815
