| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 14-s + 16-s + 20-s − 22-s − 28-s − 2·29-s − 31-s − 32-s − 35-s − 40-s + 44-s + 53-s + 55-s + 56-s + 2·58-s − 2·59-s + 62-s + 64-s + 70-s − 73-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 14-s + 16-s + 20-s − 22-s − 28-s − 2·29-s − 31-s − 32-s − 35-s − 40-s + 44-s + 53-s + 55-s + 56-s + 2·58-s − 2·59-s + 62-s + 64-s + 70-s − 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5214974113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5214974113\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−12.47396114425846307743873615259, −11.38989650496637321338860527928, −10.35992325012824794665196702722, −9.410495748316099838078174132841, −9.099393303231436904907047317976, −7.51996723259934097038513768964, −6.48867944268676963804051294529, −5.73406277885848990045189008369, −3.50170398943400689821559251320, −1.90671030140387469693051577346,
1.90671030140387469693051577346, 3.50170398943400689821559251320, 5.73406277885848990045189008369, 6.48867944268676963804051294529, 7.51996723259934097038513768964, 9.099393303231436904907047317976, 9.410495748316099838078174132841, 10.35992325012824794665196702722, 11.38989650496637321338860527928, 12.47396114425846307743873615259
