L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 5·11-s − 12-s − 6·13-s + 15-s + 16-s − 2·17-s − 18-s − 19-s − 20-s + 5·22-s − 9·23-s + 24-s − 4·25-s + 6·26-s − 27-s − 8·29-s − 30-s − 4·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.50·11-s − 0.288·12-s − 1.66·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 1.06·22-s − 1.87·23-s + 0.204·24-s − 4/5·25-s + 1.17·26-s − 0.192·27-s − 1.48·29-s − 0.182·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.62438540152217429557567321073, −6.98377443340473819256355165353, −5.91854116488776769127658410946, −5.41613343362776044962553678886, −4.56283329655407918876065364755, −3.70714819296620105741615845861, −2.47402227355164316526885695744, −1.93757111635202522501098764637, 0, 0,
1.93757111635202522501098764637, 2.47402227355164316526885695744, 3.70714819296620105741615845861, 4.56283329655407918876065364755, 5.41613343362776044962553678886, 5.91854116488776769127658410946, 6.98377443340473819256355165353, 7.62438540152217429557567321073