L(s) = 1 | + 9.38·2-s + 13.7·3-s + 56.0·4-s − 25·5-s + 129.·6-s − 129.·7-s + 225.·8-s − 53.7·9-s − 234.·10-s + 121·11-s + 770.·12-s + 848.·13-s − 1.21e3·14-s − 343.·15-s + 320.·16-s − 1.28e3·17-s − 504.·18-s + 361·19-s − 1.40e3·20-s − 1.78e3·21-s + 1.13e3·22-s + 741.·23-s + 3.09e3·24-s + 625·25-s + 7.96e3·26-s − 4.08e3·27-s − 7.25e3·28-s + ⋯ |
L(s) = 1 | + 1.65·2-s + 0.882·3-s + 1.75·4-s − 0.447·5-s + 1.46·6-s − 0.999·7-s + 1.24·8-s − 0.221·9-s − 0.741·10-s + 0.301·11-s + 1.54·12-s + 1.39·13-s − 1.65·14-s − 0.394·15-s + 0.313·16-s − 1.07·17-s − 0.366·18-s + 0.229·19-s − 0.782·20-s − 0.881·21-s + 0.500·22-s + 0.292·23-s + 1.09·24-s + 0.200·25-s + 2.31·26-s − 1.07·27-s − 1.74·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
good | 2 | \( 1 - 9.38T + 32T^{2} \) |
| 3 | \( 1 - 13.7T + 243T^{2} \) |
| 7 | \( 1 + 129.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 848.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.28e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 741.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.40e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.06e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.54e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.42e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.38e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.77e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.81e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.03e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 243.T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.74e5T + 8.58e9T^{2} \) |
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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
−8.856757700722954867888334019592, −7.78931022341129502125149141301, −6.73821997751538254327911347938, −6.19693642046277075925641682519, −5.23658640775220291434332571294, −4.02055882232573913858352017958, −3.54271431612027466989090041896, −2.89437229024764373984432258586, −1.78759664498048685052763108768, 0,
1.78759664498048685052763108768, 2.89437229024764373984432258586, 3.54271431612027466989090041896, 4.02055882232573913858352017958, 5.23658640775220291434332571294, 6.19693642046277075925641682519, 6.73821997751538254327911347938, 7.78931022341129502125149141301, 8.856757700722954867888334019592