L(s) = 1 | + 3-s + 9-s − 2·11-s − 25-s + 27-s − 2·33-s − 49-s + 2·59-s − 2·73-s − 75-s + 81-s − 2·83-s + 2·97-s − 2·99-s + 2·107-s + ⋯ |
L(s) = 1 | + 3-s + 9-s − 2·11-s − 25-s + 27-s − 2·33-s − 49-s + 2·59-s − 2·73-s − 75-s + 81-s − 2·83-s + 2·97-s − 2·99-s + 2·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.038832581\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.038832581\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 + T )^{2} \) |
| 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 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 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
−11.51594431833325755850778013775, −10.37209973055988199399442480591, −9.846246176532358425912726735900, −8.645607284808412110813643206624, −7.923175260352088101919956375328, −7.17586826676654994203448376032, −5.69772240246461967573219910016, −4.56816156008636606652577181448, −3.22873040444622756373737015810, −2.18028315741103604715303308093,
2.18028315741103604715303308093, 3.22873040444622756373737015810, 4.56816156008636606652577181448, 5.69772240246461967573219910016, 7.17586826676654994203448376032, 7.923175260352088101919956375328, 8.645607284808412110813643206624, 9.846246176532358425912726735900, 10.37209973055988199399442480591, 11.51594431833325755850778013775