| L(s) = 1 | − 3-s + 5-s + 9-s + 13-s − 15-s − 2·17-s − 4·19-s + 4·23-s + 25-s − 27-s − 4·31-s − 8·37-s − 39-s − 6·41-s − 8·43-s + 45-s + 2·47-s − 7·49-s + 2·51-s − 6·53-s + 4·57-s − 4·59-s + 2·61-s + 65-s + 2·67-s − 4·69-s − 14·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.277·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.718·31-s − 1.31·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 0.291·47-s − 49-s + 0.280·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.124·65-s + 0.244·67-s − 0.481·69-s − 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8826486346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8826486346\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.01755117501443, −12.76211267507232, −12.16703743177759, −11.69415154578081, −11.16886183274353, −10.78932642576821, −10.35918271715625, −9.937657393094158, −9.259524857223563, −8.912526425534808, −8.413357293185814, −7.884007720687141, −7.132324059523616, −6.820021648873808, −6.291479314113607, −5.934035024891180, −5.087282286903930, −4.981884536829758, −4.332776686332781, −3.524436432906188, −3.240908875800212, −2.284609993289447, −1.807119655328286, −1.248244156277236, −0.2774444542241552,
0.2774444542241552, 1.248244156277236, 1.807119655328286, 2.284609993289447, 3.240908875800212, 3.524436432906188, 4.332776686332781, 4.981884536829758, 5.087282286903930, 5.934035024891180, 6.291479314113607, 6.820021648873808, 7.132324059523616, 7.884007720687141, 8.413357293185814, 8.912526425534808, 9.259524857223563, 9.937657393094158, 10.35918271715625, 10.78932642576821, 11.16886183274353, 11.69415154578081, 12.16703743177759, 12.76211267507232, 13.01755117501443
