| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 16.3·5-s − 6·6-s + 31.0·7-s + 8·8-s + 9·9-s − 32.6·10-s + 4.34·11-s − 12·12-s − 24·13-s + 62.0·14-s + 49.0·15-s + 16·16-s − 6.42·17-s + 18·18-s − 65.3·20-s − 93.1·21-s + 8.69·22-s − 157.·23-s − 24·24-s + 142.·25-s − 48·26-s − 27·27-s + 124.·28-s + 248.·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.46·5-s − 0.408·6-s + 1.67·7-s + 0.353·8-s + 0.333·9-s − 1.03·10-s + 0.119·11-s − 0.288·12-s − 0.512·13-s + 1.18·14-s + 0.844·15-s + 0.250·16-s − 0.0916·17-s + 0.235·18-s − 0.731·20-s − 0.967·21-s + 0.0842·22-s − 1.42·23-s − 0.204·24-s + 1.13·25-s − 0.362·26-s − 0.192·27-s + 0.837·28-s + 1.58·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 16.3T + 125T^{2} \) |
| 7 | \( 1 - 31.0T + 343T^{2} \) |
| 11 | \( 1 - 4.34T + 1.33e3T^{2} \) |
| 13 | \( 1 + 24T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.42T + 4.91e3T^{2} \) |
| 23 | \( 1 + 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 248.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 308.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 84T + 5.06e4T^{2} \) |
| 41 | \( 1 - 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 481.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 426.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 184.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 584.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 291.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 420.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 628.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 197.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 519.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 992.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−8.036659681196838015135508047342, −7.58764446970922911705325275767, −6.85393051472492115952056279791, −5.72939705100885225420786713078, −4.95233707045531842515207540740, −4.33446567658764053277134213547, −3.75246077364733370776585748927, −2.36566289956847142883633111439, −1.27057637556805278464162878943, 0,
1.27057637556805278464162878943, 2.36566289956847142883633111439, 3.75246077364733370776585748927, 4.33446567658764053277134213547, 4.95233707045531842515207540740, 5.72939705100885225420786713078, 6.85393051472492115952056279791, 7.58764446970922911705325275767, 8.036659681196838015135508047342
