| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 4.06·5-s − 6·6-s + 33.2·7-s + 8·8-s + 9·9-s − 8.12·10-s + 5.11·11-s − 12·12-s − 18.3·13-s + 66.5·14-s + 12.1·15-s + 16·16-s + 18·18-s − 66.8·19-s − 16.2·20-s − 99.8·21-s + 10.2·22-s − 160.·23-s − 24·24-s − 108.·25-s − 36.6·26-s − 27·27-s + 133.·28-s − 170.·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.363·5-s − 0.408·6-s + 1.79·7-s + 0.353·8-s + 0.333·9-s − 0.257·10-s + 0.140·11-s − 0.288·12-s − 0.391·13-s + 1.27·14-s + 0.209·15-s + 0.250·16-s + 0.235·18-s − 0.806·19-s − 0.181·20-s − 1.03·21-s + 0.0990·22-s − 1.45·23-s − 0.204·24-s − 0.867·25-s − 0.276·26-s − 0.192·27-s + 0.898·28-s − 1.09·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 4.06T + 125T^{2} \) |
| 7 | \( 1 - 33.2T + 343T^{2} \) |
| 11 | \( 1 - 5.11T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 66.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 170.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 258.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 307.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 280.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 328.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 55.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 233.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 7.28T + 2.26e5T^{2} \) |
| 67 | \( 1 - 314.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 96.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 10.1T + 4.93e5T^{2} \) |
| 83 | \( 1 + 66.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 202.T + 9.12e5T^{2} \) |
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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
−8.305860086467811584715479900762, −7.65766417576811461275647391839, −7.02843706758001255614942100919, −5.73691840293550909290090512371, −5.41268423966566913533429379177, −4.25415093799734227330482264416, −3.99359322697512924971973934587, −2.23822653038903419060951625573, −1.56432268640662663815140285228, 0,
1.56432268640662663815140285228, 2.23822653038903419060951625573, 3.99359322697512924971973934587, 4.25415093799734227330482264416, 5.41268423966566913533429379177, 5.73691840293550909290090512371, 7.02843706758001255614942100919, 7.65766417576811461275647391839, 8.305860086467811584715479900762
