| L(s) = 1 | + 3-s − 0.812·5-s − 4.94·7-s + 9-s + 1.85·11-s − 6.88·13-s − 0.812·15-s − 7.62·17-s + 0.765·19-s − 4.94·21-s + 23-s − 4.34·25-s + 27-s − 29-s − 7.16·31-s + 1.85·33-s + 4.01·35-s − 6.94·37-s − 6.88·39-s − 2.06·41-s + 12.9·43-s − 0.812·45-s − 3.03·47-s + 17.4·49-s − 7.62·51-s + 13.0·53-s − 1.50·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.363·5-s − 1.86·7-s + 0.333·9-s + 0.559·11-s − 1.90·13-s − 0.209·15-s − 1.84·17-s + 0.175·19-s − 1.07·21-s + 0.208·23-s − 0.868·25-s + 0.192·27-s − 0.185·29-s − 1.28·31-s + 0.323·33-s + 0.678·35-s − 1.14·37-s − 1.10·39-s − 0.322·41-s + 1.97·43-s − 0.121·45-s − 0.443·47-s + 2.49·49-s − 1.06·51-s + 1.79·53-s − 0.203·55-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
| \(L(1)\) |
\(\approx\) |
\(0.6977347338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6977347338\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.812T + 5T^{2} \) |
| 7 | \( 1 + 4.94T + 7T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 13 | \( 1 + 6.88T + 13T^{2} \) |
| 17 | \( 1 + 7.62T + 17T^{2} \) |
| 19 | \( 1 - 0.765T + 19T^{2} \) |
| 31 | \( 1 + 7.16T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 + 2.06T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 + 3.03T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 6.09T + 59T^{2} \) |
| 61 | \( 1 + 3.49T + 61T^{2} \) |
| 67 | \( 1 + 9.90T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 3.62T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 0.720T + 83T^{2} \) |
| 89 | \( 1 - 4.99T + 89T^{2} \) |
| 97 | \( 1 + 5.22T + 97T^{2} \) |
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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
−7.60929726419638821332242434053, −7.10347375961826564706446928957, −6.71260103978835685221179108793, −5.86417379135726990973995045846, −4.94566828595683956615819503703, −4.02212550830512123224955581619, −3.60130225143883864581617015602, −2.60109442878555114956150671420, −2.13641410253276904562852837446, −0.36840260269670983551651814595,
0.36840260269670983551651814595, 2.13641410253276904562852837446, 2.60109442878555114956150671420, 3.60130225143883864581617015602, 4.02212550830512123224955581619, 4.94566828595683956615819503703, 5.86417379135726990973995045846, 6.71260103978835685221179108793, 7.10347375961826564706446928957, 7.60929726419638821332242434053
