| L(s) = 1 | + 3-s + 9-s + 13-s + 2·17-s + 27-s − 2·29-s + 4·31-s − 6·37-s + 39-s + 6·41-s − 4·43-s − 4·47-s + 2·51-s + 10·53-s + 2·61-s − 8·67-s + 4·71-s − 6·73-s − 8·79-s + 81-s + 8·83-s − 2·87-s + 6·89-s + 4·93-s − 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.277·13-s + 0.485·17-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.986·37-s + 0.160·39-s + 0.937·41-s − 0.609·43-s − 0.583·47-s + 0.280·51-s + 1.37·53-s + 0.256·61-s − 0.977·67-s + 0.474·71-s − 0.702·73-s − 0.900·79-s + 1/9·81-s + 0.878·83-s − 0.214·87-s + 0.635·89-s + 0.414·93-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.83159626832014, −12.15866106526172, −11.89576055532876, −11.38848512882588, −10.82415058699710, −10.37645498605839, −10.00442244445430, −9.515989389029518, −9.046235804224544, −8.550904877456569, −8.264688716518423, −7.636532536758653, −7.303142643352998, −6.766092444409814, −6.254397062067281, −5.738031052559302, −5.227857492192932, −4.703844037401849, −4.115602098550054, −3.672506913492143, −3.147648187368886, −2.640461154704095, −2.042984514706722, −1.437396376884809, −0.8674627548717346, 0,
0.8674627548717346, 1.437396376884809, 2.042984514706722, 2.640461154704095, 3.147648187368886, 3.672506913492143, 4.115602098550054, 4.703844037401849, 5.227857492192932, 5.738031052559302, 6.254397062067281, 6.766092444409814, 7.303142643352998, 7.636532536758653, 8.264688716518423, 8.550904877456569, 9.046235804224544, 9.515989389029518, 10.00442244445430, 10.37645498605839, 10.82415058699710, 11.38848512882588, 11.89576055532876, 12.15866106526172, 12.83159626832014
