| L(s) = 1 | − 0.648·3-s + 1.55·5-s + 3.00·7-s − 2.57·9-s − 11-s − 2.37·13-s − 1.00·15-s − 0.984·17-s + 19-s − 1.95·21-s − 3.60·23-s − 2.57·25-s + 3.62·27-s + 7.28·29-s + 10.1·31-s + 0.648·33-s + 4.68·35-s − 6.17·37-s + 1.54·39-s + 0.843·41-s + 9.86·43-s − 4.01·45-s + 13.7·47-s + 2.05·49-s + 0.638·51-s + 2.54·53-s − 1.55·55-s + ⋯ |
| L(s) = 1 | − 0.374·3-s + 0.695·5-s + 1.13·7-s − 0.859·9-s − 0.301·11-s − 0.659·13-s − 0.260·15-s − 0.238·17-s + 0.229·19-s − 0.426·21-s − 0.751·23-s − 0.515·25-s + 0.696·27-s + 1.35·29-s + 1.83·31-s + 0.112·33-s + 0.791·35-s − 1.01·37-s + 0.247·39-s + 0.131·41-s + 1.50·43-s − 0.598·45-s + 1.99·47-s + 0.293·49-s + 0.0894·51-s + 0.350·53-s − 0.209·55-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
| \(L(1)\) |
\(\approx\) |
\(1.864051894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.864051894\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.648T + 3T^{2} \) |
| 5 | \( 1 - 1.55T + 5T^{2} \) |
| 7 | \( 1 - 3.00T + 7T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 17 | \( 1 + 0.984T + 17T^{2} \) |
| 23 | \( 1 + 3.60T + 23T^{2} \) |
| 29 | \( 1 - 7.28T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 - 0.843T + 41T^{2} \) |
| 43 | \( 1 - 9.86T + 43T^{2} \) |
| 47 | \( 1 - 13.7T + 47T^{2} \) |
| 53 | \( 1 - 2.54T + 53T^{2} \) |
| 59 | \( 1 + 9.68T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 9.22T + 71T^{2} \) |
| 73 | \( 1 + 3.67T + 73T^{2} \) |
| 79 | \( 1 - 4.14T + 79T^{2} \) |
| 83 | \( 1 + 4.87T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 9.28T + 97T^{2} \) |
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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
−8.444628315577461076490174774271, −8.066915915875998207982170012492, −7.11443312930071368626424043029, −6.21421135048346044829826602320, −5.56320585592879475414349799595, −4.94448854128579280843347800506, −4.16549966795268995290795807357, −2.73844885377036331350869085990, −2.14158926645011181949343786268, −0.830880227302575651546588562832,
0.830880227302575651546588562832, 2.14158926645011181949343786268, 2.73844885377036331350869085990, 4.16549966795268995290795807357, 4.94448854128579280843347800506, 5.56320585592879475414349799595, 6.21421135048346044829826602320, 7.11443312930071368626424043029, 8.066915915875998207982170012492, 8.444628315577461076490174774271
