| L(s) = 1 | − 2·2-s + 8.18·3-s + 4·4-s − 0.929·5-s − 16.3·6-s − 8·8-s + 40.0·9-s + 1.85·10-s + 24.7·11-s + 32.7·12-s − 49.9·13-s − 7.61·15-s + 16·16-s − 47.0·17-s − 80.0·18-s + 28.1·19-s − 3.71·20-s − 49.5·22-s + 23·23-s − 65.4·24-s − 124.·25-s + 99.9·26-s + 106.·27-s − 83.2·29-s + 15.2·30-s + 104.·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.57·3-s + 0.5·4-s − 0.0831·5-s − 1.11·6-s − 0.353·8-s + 1.48·9-s + 0.0588·10-s + 0.679·11-s + 0.787·12-s − 1.06·13-s − 0.131·15-s + 0.250·16-s − 0.671·17-s − 1.04·18-s + 0.340·19-s − 0.0415·20-s − 0.480·22-s + 0.208·23-s − 0.557·24-s − 0.993·25-s + 0.753·26-s + 0.760·27-s − 0.533·29-s + 0.0926·30-s + 0.603·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 8.18T + 27T^{2} \) |
| 5 | \( 1 + 0.929T + 125T^{2} \) |
| 11 | \( 1 - 24.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 28.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 83.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 104.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 131.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 214.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 0.790T + 7.95e4T^{2} \) |
| 47 | \( 1 - 45.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 643.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 368.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 82.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 47.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 321.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 301.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 648.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 267.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 27.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 233.T + 9.12e5T^{2} \) |
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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
−8.381805012890302862020378692875, −7.67591931759141193450089761006, −7.12949052883476370369737499655, −6.25383663885034109095420892985, −4.93887517143472173969883520230, −3.94408239201415906885395128112, −3.12113021789760028680137735486, −2.26789688440956987972640254751, −1.50827829947494103147214638027, 0,
1.50827829947494103147214638027, 2.26789688440956987972640254751, 3.12113021789760028680137735486, 3.94408239201415906885395128112, 4.93887517143472173969883520230, 6.25383663885034109095420892985, 7.12949052883476370369737499655, 7.67591931759141193450089761006, 8.381805012890302862020378692875
