| L(s) = 1 | + 3-s − 4-s + 7-s + 9-s − 12-s − 2·13-s + 16-s + 19-s + 21-s − 25-s + 27-s − 28-s − 2·31-s − 36-s − 2·39-s − 2·43-s + 48-s + 49-s + 2·52-s + 57-s + 63-s − 64-s − 75-s − 76-s + 81-s − 84-s − 2·91-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s + 7-s + 9-s − 12-s − 2·13-s + 16-s + 19-s + 21-s − 25-s + 27-s − 28-s − 2·31-s − 36-s − 2·39-s − 2·43-s + 48-s + 49-s + 2·52-s + 57-s + 63-s − 64-s − 75-s − 76-s + 81-s − 84-s − 2·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9700189432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9700189432\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−11.64306933003714186349319097229, −10.20568624685785553513996155681, −9.598887409507898166913745997564, −8.803829250796251504034054783454, −7.79783712035478158965812803741, −7.32575159115761954413284756626, −5.32442223728901045494264294612, −4.63709153013307881910966219517, −3.46100772776276641971135054557, −1.95205145151133716131073489496,
1.95205145151133716131073489496, 3.46100772776276641971135054557, 4.63709153013307881910966219517, 5.32442223728901045494264294612, 7.32575159115761954413284756626, 7.79783712035478158965812803741, 8.803829250796251504034054783454, 9.598887409507898166913745997564, 10.20568624685785553513996155681, 11.64306933003714186349319097229
