L(s) = 1 | + 1.56·3-s − 2.37·5-s + 2.65·7-s − 0.557·9-s − 2.81·11-s + 2.78·13-s − 3.71·15-s − 7.35·17-s + 19-s + 4.15·21-s + 5.90·23-s + 0.642·25-s − 5.55·27-s + 2.21·29-s + 1.03·31-s − 4.39·33-s − 6.31·35-s + 4.56·37-s + 4.36·39-s + 2.65·41-s + 6.23·43-s + 1.32·45-s + 0.283·47-s + 0.0658·49-s − 11.4·51-s − 8.74·53-s + 6.68·55-s + ⋯ |
L(s) = 1 | + 0.902·3-s − 1.06·5-s + 1.00·7-s − 0.185·9-s − 0.847·11-s + 0.773·13-s − 0.958·15-s − 1.78·17-s + 0.229·19-s + 0.906·21-s + 1.23·23-s + 0.128·25-s − 1.07·27-s + 0.411·29-s + 0.185·31-s − 0.765·33-s − 1.06·35-s + 0.750·37-s + 0.698·39-s + 0.414·41-s + 0.951·43-s + 0.197·45-s + 0.0413·47-s + 0.00940·49-s − 1.60·51-s − 1.20·53-s + 0.900·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 2.65T + 7T^{2} \) |
| 11 | \( 1 + 2.81T + 11T^{2} \) |
| 13 | \( 1 - 2.78T + 13T^{2} \) |
| 17 | \( 1 + 7.35T + 17T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 - 2.21T + 29T^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 - 4.56T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 - 0.283T + 47T^{2} \) |
| 53 | \( 1 + 8.74T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 2.64T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 0.602T + 71T^{2} \) |
| 73 | \( 1 + 4.37T + 73T^{2} \) |
| 83 | \( 1 + 2.88T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.86476020214158626292067026843, −7.34835949806269485757130555346, −6.41386792425512732114392676301, −5.47068311688547321295190019084, −4.54958116295088684230550486657, −4.15624928206801817502538946543, −3.07997383879319506901666839838, −2.54816457309780041194297468909, −1.42853497338071704016563511861, 0,
1.42853497338071704016563511861, 2.54816457309780041194297468909, 3.07997383879319506901666839838, 4.15624928206801817502538946543, 4.54958116295088684230550486657, 5.47068311688547321295190019084, 6.41386792425512732114392676301, 7.34835949806269485757130555346, 7.86476020214158626292067026843