L(s) = 1 | + 7-s − 9-s − 2·23-s − 25-s + 49-s − 63-s + 2·71-s + 2·79-s + 81-s − 2·113-s + ⋯ |
L(s) = 1 | + 7-s − 9-s − 2·23-s − 25-s + 49-s − 63-s + 2·71-s + 2·79-s + 81-s − 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7266038497\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7266038497\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 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
−12.22369903725000176271682439712, −11.58836941877609302991849398627, −10.69844401423555597352562836766, −9.562484600326469358309217586395, −8.370993614097800613301780281956, −7.76810251622520307131534426357, −6.22032254060822648294042949415, −5.24138774305761413651259165297, −3.89328530717634633460037294551, −2.16883543425635289528621359563,
2.16883543425635289528621359563, 3.89328530717634633460037294551, 5.24138774305761413651259165297, 6.22032254060822648294042949415, 7.76810251622520307131534426357, 8.370993614097800613301780281956, 9.562484600326469358309217586395, 10.69844401423555597352562836766, 11.58836941877609302991849398627, 12.22369903725000176271682439712