| L(s) = 1 | − 3-s − 4-s + 5-s + 9-s + 12-s − 15-s + 16-s − 20-s − 2·23-s + 25-s − 27-s + 2·31-s − 36-s + 45-s + 2·47-s − 48-s + 49-s + 2·53-s + 60-s − 64-s + 2·69-s − 75-s + 80-s + 81-s + 2·92-s − 2·93-s − 100-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s + 5-s + 9-s + 12-s − 15-s + 16-s − 20-s − 2·23-s + 25-s − 27-s + 2·31-s − 36-s + 45-s + 2·47-s − 48-s + 49-s + 2·53-s + 60-s − 64-s + 2·69-s − 75-s + 80-s + 81-s + 2·92-s − 2·93-s − 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7851425395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7851425395\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.649207635616483733348189541693, −8.797449252363717347477788552082, −7.948910075081726405946026308930, −6.90259195084070242024747029966, −5.94240985892354588055328645653, −5.60223294448136324263401801227, −4.60098807297338012289901511506, −3.94118391816636433464815226977, −2.35086277687952367790681256445, −0.988528584807439971347255214798,
0.988528584807439971347255214798, 2.35086277687952367790681256445, 3.94118391816636433464815226977, 4.60098807297338012289901511506, 5.60223294448136324263401801227, 5.94240985892354588055328645653, 6.90259195084070242024747029966, 7.948910075081726405946026308930, 8.797449252363717347477788552082, 9.649207635616483733348189541693
