| L(s) = 1 | − 1.34·3-s + 3.97·5-s + 7·7-s − 25.2·9-s + 11·11-s + 26.5·13-s − 5.33·15-s − 90.7·17-s + 68.4·19-s − 9.39·21-s + 36.3·23-s − 109.·25-s + 70.0·27-s + 176.·29-s − 177.·31-s − 14.7·33-s + 27.8·35-s − 299.·37-s − 35.6·39-s − 45.3·41-s + 528.·43-s − 100.·45-s − 357.·47-s + 49·49-s + 121.·51-s − 742.·53-s + 43.7·55-s + ⋯ |
| L(s) = 1 | − 0.258·3-s + 0.355·5-s + 0.377·7-s − 0.933·9-s + 0.301·11-s + 0.566·13-s − 0.0918·15-s − 1.29·17-s + 0.825·19-s − 0.0975·21-s + 0.329·23-s − 0.873·25-s + 0.499·27-s + 1.12·29-s − 1.02·31-s − 0.0778·33-s + 0.134·35-s − 1.32·37-s − 0.146·39-s − 0.172·41-s + 1.87·43-s − 0.331·45-s − 1.10·47-s + 0.142·49-s + 0.334·51-s − 1.92·53-s + 0.107·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 1.34T + 27T^{2} \) |
| 5 | \( 1 - 3.97T + 125T^{2} \) |
| 13 | \( 1 - 26.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 90.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 68.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 36.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 176.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 299.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 45.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 528.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 357.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 742.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 877.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 199.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 998.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 27.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 161.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 624.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 396.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.40e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 193.T + 9.12e5T^{2} \) |
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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
−8.886503544550114050148766949211, −8.293128098734129805926615470389, −7.20599644483045384026345668609, −6.33472297247191506004952394664, −5.60150641882922079421586460391, −4.75589468995490599786438338057, −3.63902757598236937799495658977, −2.53078046987111218618388731687, −1.39158271378551134143670489421, 0,
1.39158271378551134143670489421, 2.53078046987111218618388731687, 3.63902757598236937799495658977, 4.75589468995490599786438338057, 5.60150641882922079421586460391, 6.33472297247191506004952394664, 7.20599644483045384026345668609, 8.293128098734129805926615470389, 8.886503544550114050148766949211
