| L(s) = 1 | − 2.32·3-s − 5-s + 0.515·7-s + 2.38·9-s + 4.54·11-s + 1.42·13-s + 2.32·15-s + 3.21·17-s − 2.56·19-s − 1.19·21-s − 0.925·23-s + 25-s + 1.41·27-s − 1.44·29-s − 6.16·31-s − 10.5·33-s − 0.515·35-s − 5.88·37-s − 3.29·39-s − 9.02·41-s − 7.54·43-s − 2.38·45-s − 1.06·47-s − 6.73·49-s − 7.46·51-s + 3.39·53-s − 4.54·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s − 0.447·5-s + 0.194·7-s + 0.796·9-s + 1.36·11-s + 0.394·13-s + 0.599·15-s + 0.779·17-s − 0.589·19-s − 0.261·21-s − 0.192·23-s + 0.200·25-s + 0.272·27-s − 0.268·29-s − 1.10·31-s − 1.83·33-s − 0.0871·35-s − 0.966·37-s − 0.528·39-s − 1.40·41-s − 1.15·43-s − 0.356·45-s − 0.156·47-s − 0.961·49-s − 1.04·51-s + 0.466·53-s − 0.612·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 2.32T + 3T^{2} \) |
| 7 | \( 1 - 0.515T + 7T^{2} \) |
| 11 | \( 1 - 4.54T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 + 0.925T + 23T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 + 6.16T + 31T^{2} \) |
| 37 | \( 1 + 5.88T + 37T^{2} \) |
| 41 | \( 1 + 9.02T + 41T^{2} \) |
| 43 | \( 1 + 7.54T + 43T^{2} \) |
| 47 | \( 1 + 1.06T + 47T^{2} \) |
| 53 | \( 1 - 3.39T + 53T^{2} \) |
| 59 | \( 1 - 8.53T + 59T^{2} \) |
| 61 | \( 1 - 4.29T + 61T^{2} \) |
| 67 | \( 1 + 7.75T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 - 6.18T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 7.67T + 97T^{2} \) |
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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
−7.27182227478616663748261108834, −6.60779662100161509432794708672, −6.24208981359304027120963945942, −5.30068427253479100530005306911, −4.91610045119244416998237359464, −3.83287299330737000233476214654, −3.49328264546655901632436313389, −1.91837407689772630585300363021, −1.09247427952807017573011427261, 0,
1.09247427952807017573011427261, 1.91837407689772630585300363021, 3.49328264546655901632436313389, 3.83287299330737000233476214654, 4.91610045119244416998237359464, 5.30068427253479100530005306911, 6.24208981359304027120963945942, 6.60779662100161509432794708672, 7.27182227478616663748261108834
