| L(s) = 1 | − 3-s − 3.33·5-s − 2.20·7-s + 9-s + 4.06·11-s + 6.44·13-s + 3.33·15-s + 6.35·17-s + 6.75·19-s + 2.20·21-s + 23-s + 6.15·25-s − 27-s + 29-s + 3.50·31-s − 4.06·33-s + 7.37·35-s + 1.43·37-s − 6.44·39-s + 10.9·41-s − 11.9·43-s − 3.33·45-s − 5.99·47-s − 2.11·49-s − 6.35·51-s − 4.14·53-s − 13.5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.49·5-s − 0.835·7-s + 0.333·9-s + 1.22·11-s + 1.78·13-s + 0.862·15-s + 1.54·17-s + 1.54·19-s + 0.482·21-s + 0.208·23-s + 1.23·25-s − 0.192·27-s + 0.185·29-s + 0.630·31-s − 0.707·33-s + 1.24·35-s + 0.235·37-s − 1.03·39-s + 1.71·41-s − 1.81·43-s − 0.497·45-s − 0.874·47-s − 0.302·49-s − 0.889·51-s − 0.569·53-s − 1.83·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.551411512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551411512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 3.33T + 5T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 - 6.44T + 13T^{2} \) |
| 17 | \( 1 - 6.35T + 17T^{2} \) |
| 19 | \( 1 - 6.75T + 19T^{2} \) |
| 31 | \( 1 - 3.50T + 31T^{2} \) |
| 37 | \( 1 - 1.43T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 5.99T + 47T^{2} \) |
| 53 | \( 1 + 4.14T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 2.00T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 + 6.38T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 6.82T + 89T^{2} \) |
| 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
−7.957919483398108284659180539683, −7.00795787825232140639775782294, −6.53589662456220126886256118283, −5.85336110860989664487703601515, −5.05031073490407217617408260892, −4.05255856118469376769820223978, −3.53337350632378209281943114981, −3.18174457799133408198328838991, −1.26614206233050415912771756557, −0.76975637896183821871004633539,
0.76975637896183821871004633539, 1.26614206233050415912771756557, 3.18174457799133408198328838991, 3.53337350632378209281943114981, 4.05255856118469376769820223978, 5.05031073490407217617408260892, 5.85336110860989664487703601515, 6.53589662456220126886256118283, 7.00795787825232140639775782294, 7.957919483398108284659180539683
