| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2.73·7-s − 8-s + 9-s + 0.267·11-s − 12-s + 0.732·13-s + 2.73·14-s + 16-s + 4.19·17-s − 18-s − 19-s + 2.73·21-s − 0.267·22-s − 7.92·23-s + 24-s − 0.732·26-s − 27-s − 2.73·28-s + 1.73·29-s + 4.46·31-s − 32-s − 0.267·33-s − 4.19·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.03·7-s − 0.353·8-s + 0.333·9-s + 0.0807·11-s − 0.288·12-s + 0.203·13-s + 0.730·14-s + 0.250·16-s + 1.01·17-s − 0.235·18-s − 0.229·19-s + 0.596·21-s − 0.0571·22-s − 1.65·23-s + 0.204·24-s − 0.143·26-s − 0.192·27-s − 0.516·28-s + 0.321·29-s + 0.801·31-s − 0.176·32-s − 0.0466·33-s − 0.719·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 - 0.267T + 11T^{2} \) |
| 13 | \( 1 - 0.732T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 23 | \( 1 + 7.92T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 1.73T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 + 6.66T + 61T^{2} \) |
| 67 | \( 1 + 3.73T + 67T^{2} \) |
| 71 | \( 1 - 1.80T + 71T^{2} \) |
| 73 | \( 1 - 4.46T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 0.267T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 9.12T + 97T^{2} \) |
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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
−8.360220202259788747413491715663, −7.70982633546656140732920970351, −6.86463295501935493794246110714, −6.12746734437435725474151843130, −5.69493695338968508555443379806, −4.41123081895737540513744296132, −3.50897135866188363673771118030, −2.51491091965069771385375490574, −1.22005894131790457823118301979, 0,
1.22005894131790457823118301979, 2.51491091965069771385375490574, 3.50897135866188363673771118030, 4.41123081895737540513744296132, 5.69493695338968508555443379806, 6.12746734437435725474151843130, 6.86463295501935493794246110714, 7.70982633546656140732920970351, 8.360220202259788747413491715663
