| L(s) = 1 | + 1.71·2-s + 1.60·3-s + 0.933·4-s + 5-s + 2.75·6-s + 7-s − 1.82·8-s − 0.420·9-s + 1.71·10-s − 2.49·11-s + 1.49·12-s − 0.606·13-s + 1.71·14-s + 1.60·15-s − 4.99·16-s − 2.88·17-s − 0.719·18-s + 0.944·19-s + 0.933·20-s + 1.60·21-s − 4.27·22-s − 6.49·23-s − 2.93·24-s + 25-s − 1.03·26-s − 5.49·27-s + 0.933·28-s + ⋯ |
| L(s) = 1 | + 1.21·2-s + 0.927·3-s + 0.466·4-s + 0.447·5-s + 1.12·6-s + 0.377·7-s − 0.646·8-s − 0.140·9-s + 0.541·10-s − 0.753·11-s + 0.432·12-s − 0.168·13-s + 0.457·14-s + 0.414·15-s − 1.24·16-s − 0.699·17-s − 0.169·18-s + 0.216·19-s + 0.208·20-s + 0.350·21-s − 0.912·22-s − 1.35·23-s − 0.599·24-s + 0.200·25-s − 0.203·26-s − 1.05·27-s + 0.176·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 1.71T + 2T^{2} \) |
| 3 | \( 1 - 1.60T + 3T^{2} \) |
| 11 | \( 1 + 2.49T + 11T^{2} \) |
| 13 | \( 1 + 0.606T + 13T^{2} \) |
| 17 | \( 1 + 2.88T + 17T^{2} \) |
| 19 | \( 1 - 0.944T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 + 0.815T + 29T^{2} \) |
| 31 | \( 1 - 0.429T + 31T^{2} \) |
| 37 | \( 1 + 0.337T + 37T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 - 0.933T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 3.41T + 53T^{2} \) |
| 59 | \( 1 - 3.26T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 2.87T + 67T^{2} \) |
| 71 | \( 1 + 4.64T + 71T^{2} \) |
| 73 | \( 1 + 4.55T + 73T^{2} \) |
| 79 | \( 1 + 9.80T + 79T^{2} \) |
| 83 | \( 1 - 6.09T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 3.18T + 97T^{2} \) |
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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
−7.56589017928203680542819714814, −6.60260496639894525082689114582, −5.84848476818082776282998072053, −5.40433009732572906135998224112, −4.52487264552480962938842639936, −4.00268317012650956830389383946, −3.04631128606600931593885781293, −2.55839370369180728884556659208, −1.80440512946883771721282904642, 0,
1.80440512946883771721282904642, 2.55839370369180728884556659208, 3.04631128606600931593885781293, 4.00268317012650956830389383946, 4.52487264552480962938842639936, 5.40433009732572906135998224112, 5.84848476818082776282998072053, 6.60260496639894525082689114582, 7.56589017928203680542819714814
