| L(s) = 1 | + 3·3-s + 2·7-s + 9·9-s + 22·13-s − 26·19-s + 6·21-s + 25·25-s + 27·27-s − 46·31-s − 26·37-s + 66·39-s + 22·43-s − 45·49-s − 78·57-s − 74·61-s + 18·63-s − 122·67-s − 46·73-s + 75·75-s − 142·79-s + 81·81-s + 44·91-s − 138·93-s + 2·97-s + 194·103-s + 214·109-s − 78·111-s + ⋯ |
| L(s) = 1 | + 3-s + 2/7·7-s + 9-s + 1.69·13-s − 1.36·19-s + 2/7·21-s + 25-s + 27-s − 1.48·31-s − 0.702·37-s + 1.69·39-s + 0.511·43-s − 0.918·49-s − 1.36·57-s − 1.21·61-s + 2/7·63-s − 1.82·67-s − 0.630·73-s + 75-s − 1.79·79-s + 81-s + 0.483·91-s − 1.48·93-s + 2/97·97-s + 1.88·103-s + 1.96·109-s − 0.702·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.208326295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208326295\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| good | 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 - 2 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 22 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 26 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 46 T + p^{2} T^{2} \) |
| 37 | \( 1 + 26 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 22 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 74 T + p^{2} T^{2} \) |
| 67 | \( 1 + 122 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 46 T + p^{2} T^{2} \) |
| 79 | \( 1 + 142 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 2 T + p^{2} T^{2} \) |
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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
−12.61473187465100986317566421057, −11.13112602429135677172967195379, −10.39382314282203396087111061641, −8.959671153175178664032384476176, −8.524206686848126024137066396593, −7.31461290749082860154761378751, −6.11540748883271152272567997396, −4.44365225141518026083351121884, −3.30800484609230149778015279707, −1.67565795327991440860324882684,
1.67565795327991440860324882684, 3.30800484609230149778015279707, 4.44365225141518026083351121884, 6.11540748883271152272567997396, 7.31461290749082860154761378751, 8.524206686848126024137066396593, 8.959671153175178664032384476176, 10.39382314282203396087111061641, 11.13112602429135677172967195379, 12.61473187465100986317566421057
