L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 11-s + 15-s + 17-s − 3·19-s − 2·21-s − 4·23-s + 25-s − 27-s − 7·31-s + 33-s − 2·35-s − 37-s − 4·41-s − 7·43-s − 45-s + 10·47-s − 3·49-s − 51-s − 14·53-s + 55-s + 3·57-s − 10·59-s + 5·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.258·15-s + 0.242·17-s − 0.688·19-s − 0.436·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.25·31-s + 0.174·33-s − 0.338·35-s − 0.164·37-s − 0.624·41-s − 1.06·43-s − 0.149·45-s + 1.45·47-s − 3/7·49-s − 0.140·51-s − 1.92·53-s + 0.134·55-s + 0.397·57-s − 1.30·59-s + 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3886364501\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3886364501\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.48303315736460, −12.09458943759898, −11.63555132210038, −11.19974825369028, −10.73301237759351, −10.57218028786350, −9.781896552279834, −9.542906424512497, −8.781016555458093, −8.370611657934177, −7.978905841481642, −7.509664080662307, −7.033930304185581, −6.542678154885545, −5.927692397347651, −5.566840561198702, −4.986694504907572, −4.544474006751217, −4.140165573345140, −3.464390665218762, −3.033780647374840, −2.075900572266629, −1.793062040216267, −1.097955049761960, −0.1767599896457599,
0.1767599896457599, 1.097955049761960, 1.793062040216267, 2.075900572266629, 3.033780647374840, 3.464390665218762, 4.140165573345140, 4.544474006751217, 4.986694504907572, 5.566840561198702, 5.927692397347651, 6.542678154885545, 7.033930304185581, 7.509664080662307, 7.978905841481642, 8.370611657934177, 8.781016555458093, 9.542906424512497, 9.781896552279834, 10.57218028786350, 10.73301237759351, 11.19974825369028, 11.63555132210038, 12.09458943759898, 12.48303315736460