| L(s) = 1 | − 8.72·3-s − 10.8·5-s − 35.3·7-s + 49.1·9-s − 15.0·11-s − 63.2·13-s + 94.7·15-s + 89.5·17-s − 150.·19-s + 308.·21-s − 91.3·23-s − 7.08·25-s − 193.·27-s + 29·29-s + 4.11·31-s + 131·33-s + 383.·35-s + 59.1·37-s + 551.·39-s − 346.·41-s + 483.·43-s − 533.·45-s − 75.7·47-s + 904.·49-s − 781.·51-s + 191.·53-s + 162.·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s − 0.971·5-s − 1.90·7-s + 1.82·9-s − 0.411·11-s − 1.34·13-s + 1.63·15-s + 1.27·17-s − 1.81·19-s + 3.20·21-s − 0.828·23-s − 0.0566·25-s − 1.37·27-s + 0.185·29-s + 0.0238·31-s + 0.691·33-s + 1.85·35-s + 0.262·37-s + 2.26·39-s − 1.31·41-s + 1.71·43-s − 1.76·45-s − 0.235·47-s + 2.63·49-s − 2.14·51-s + 0.496·53-s + 0.399·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 8.72T + 27T^{2} \) |
| 5 | \( 1 + 10.8T + 125T^{2} \) |
| 7 | \( 1 + 35.3T + 343T^{2} \) |
| 11 | \( 1 + 15.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 63.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 89.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 150.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 91.3T + 1.21e4T^{2} \) |
| 31 | \( 1 - 4.11T + 2.97e4T^{2} \) |
| 37 | \( 1 - 59.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 346.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 483.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 75.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 191.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 774.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 277.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 8.54T + 3.00e5T^{2} \) |
| 71 | \( 1 - 72.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 775.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 694.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 483.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 933.T + 9.12e5T^{2} \) |
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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
−8.374807725754744940823889862892, −7.37206238754356713078139169071, −6.82867298013549355281915336734, −6.06457567169303041297385744128, −5.43309591710567752402556789022, −4.37170830032869974697837495059, −3.67573718701087325263417851563, −2.45935484756113018704833355065, −0.57160965216574376756402116176, 0,
0.57160965216574376756402116176, 2.45935484756113018704833355065, 3.67573718701087325263417851563, 4.37170830032869974697837495059, 5.43309591710567752402556789022, 6.06457567169303041297385744128, 6.82867298013549355281915336734, 7.37206238754356713078139169071, 8.374807725754744940823889862892
