| L(s) = 1 | − 2.30·3-s − 2.30·5-s + 1.30·7-s + 2.30·9-s + 11-s − 5.30·13-s + 5.30·15-s + 4.60·17-s + 19-s − 3·21-s − 2.60·23-s + 0.302·25-s + 1.60·27-s − 5.69·29-s + 7.90·31-s − 2.30·33-s − 3·35-s + 8·37-s + 12.2·39-s − 7.30·41-s + 2.30·43-s − 5.30·45-s + 11.2·47-s − 5.30·49-s − 10.6·51-s + 6.60·53-s − 2.30·55-s + ⋯ |
| L(s) = 1 | − 1.32·3-s − 1.02·5-s + 0.492·7-s + 0.767·9-s + 0.301·11-s − 1.47·13-s + 1.36·15-s + 1.11·17-s + 0.229·19-s − 0.654·21-s − 0.543·23-s + 0.0605·25-s + 0.308·27-s − 1.05·29-s + 1.42·31-s − 0.400·33-s − 0.507·35-s + 1.31·37-s + 1.95·39-s − 1.14·41-s + 0.351·43-s − 0.790·45-s + 1.63·47-s − 0.757·49-s − 1.48·51-s + 0.907·53-s − 0.310·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 + 5.69T + 29T^{2} \) |
| 31 | \( 1 - 7.90T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 7.30T + 41T^{2} \) |
| 43 | \( 1 - 2.30T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 6.60T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 9.39T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−7.83200400972601025979297758630, −7.67019693490022117680034540987, −6.74562287064492887547690602924, −5.88375961149779338106889285144, −5.18609395933032407836473225252, −4.53935269492226908953650581988, −3.75051428546192640425754510789, −2.53341398825740151331511385314, −1.07731290027739869936185854387, 0,
1.07731290027739869936185854387, 2.53341398825740151331511385314, 3.75051428546192640425754510789, 4.53935269492226908953650581988, 5.18609395933032407836473225252, 5.88375961149779338106889285144, 6.74562287064492887547690602924, 7.67019693490022117680034540987, 7.83200400972601025979297758630
