| L(s) = 1 | + 1.89·2-s − 9·3-s − 28.4·4-s − 17.0·6-s − 191.·7-s − 114.·8-s + 81·9-s − 121·11-s + 255.·12-s − 536.·13-s − 363.·14-s + 692.·16-s + 1.32e3·17-s + 153.·18-s − 912.·19-s + 1.72e3·21-s − 229.·22-s + 4.12e3·23-s + 1.02e3·24-s − 1.01e3·26-s − 729·27-s + 5.45e3·28-s + 340.·29-s + 4.69e3·31-s + 4.97e3·32-s + 1.08e3·33-s + 2.50e3·34-s + ⋯ |
| L(s) = 1 | + 0.334·2-s − 0.577·3-s − 0.887·4-s − 0.193·6-s − 1.48·7-s − 0.632·8-s + 0.333·9-s − 0.301·11-s + 0.512·12-s − 0.881·13-s − 0.495·14-s + 0.676·16-s + 1.11·17-s + 0.111·18-s − 0.579·19-s + 0.855·21-s − 0.100·22-s + 1.62·23-s + 0.364·24-s − 0.295·26-s − 0.192·27-s + 1.31·28-s + 0.0751·29-s + 0.877·31-s + 0.858·32-s + 0.174·33-s + 0.372·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 1.89T + 32T^{2} \) |
| 7 | \( 1 + 191.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 536.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.32e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 912.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 340.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.27e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.74e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.94e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 628.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.66e5T + 8.58e9T^{2} \) |
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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
−9.281798954178376583905979059943, −8.262510149445065265282988805633, −7.12349563012460900721351698714, −6.32900657961925058243923931505, −5.41418205589770687612286731756, −4.70628387330088656282269961751, −3.56178820818584426333895499497, −2.78006913256440998060118264838, −0.874142811145995254770455681417, 0,
0.874142811145995254770455681417, 2.78006913256440998060118264838, 3.56178820818584426333895499497, 4.70628387330088656282269961751, 5.41418205589770687612286731756, 6.32900657961925058243923931505, 7.12349563012460900721351698714, 8.262510149445065265282988805633, 9.281798954178376583905979059943
