| L(s) = 1 | + 0.209·2-s + 3-s − 1.95·4-s + 0.209·6-s − 0.591·7-s − 0.827·8-s + 9-s − 0.870·11-s − 1.95·12-s − 1.15·13-s − 0.123·14-s + 3.73·16-s + 4.93·17-s + 0.209·18-s − 2.84·19-s − 0.591·21-s − 0.182·22-s − 6.91·23-s − 0.827·24-s − 0.240·26-s + 27-s + 1.15·28-s − 7.48·29-s + 3.45·31-s + 2.43·32-s − 0.870·33-s + 1.03·34-s + ⋯ |
| L(s) = 1 | + 0.147·2-s + 0.577·3-s − 0.978·4-s + 0.0853·6-s − 0.223·7-s − 0.292·8-s + 0.333·9-s − 0.262·11-s − 0.564·12-s − 0.319·13-s − 0.0330·14-s + 0.934·16-s + 1.19·17-s + 0.0492·18-s − 0.653·19-s − 0.128·21-s − 0.0388·22-s − 1.44·23-s − 0.168·24-s − 0.0471·26-s + 0.192·27-s + 0.218·28-s − 1.39·29-s + 0.621·31-s + 0.430·32-s − 0.151·33-s + 0.176·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.209T + 2T^{2} \) |
| 7 | \( 1 + 0.591T + 7T^{2} \) |
| 11 | \( 1 + 0.870T + 11T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 + 2.84T + 19T^{2} \) |
| 23 | \( 1 + 6.91T + 23T^{2} \) |
| 29 | \( 1 + 7.48T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 9.11T + 41T^{2} \) |
| 43 | \( 1 - 2.81T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 + 3.87T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 7.60T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 2.98T + 73T^{2} \) |
| 79 | \( 1 - 3.33T + 79T^{2} \) |
| 83 | \( 1 - 9.73T + 83T^{2} \) |
| 89 | \( 1 - 0.645T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.901688877228591264364502554278, −7.980360965887272952763674887469, −7.63477704886059674865191072043, −6.31781397284997921234492339817, −5.54354549028654799920121937230, −4.62159502210878517638389378135, −3.79771701275218674543732389836, −3.02169601939295454506829010473, −1.68715887675010045552248791215, 0,
1.68715887675010045552248791215, 3.02169601939295454506829010473, 3.79771701275218674543732389836, 4.62159502210878517638389378135, 5.54354549028654799920121937230, 6.31781397284997921234492339817, 7.63477704886059674865191072043, 7.980360965887272952763674887469, 8.901688877228591264364502554278
