L(s) = 1 | + 3-s + 9-s − 13-s − 25-s + 27-s − 39-s − 2·43-s + 49-s − 2·61-s − 75-s + 2·79-s + 81-s − 2·103-s − 117-s + ⋯ |
L(s) = 1 | + 3-s + 9-s − 13-s − 25-s + 27-s − 39-s − 2·43-s + 49-s − 2·61-s − 75-s + 2·79-s + 81-s − 2·103-s − 117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.225816668\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.225816668\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 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 | \( 1 + T^{2} \) |
| 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
−10.58854000030411290247536749754, −9.814424254235537691555677314700, −9.144904734611384843436817893333, −8.148673500168146739590215078163, −7.47452512277443153862657270921, −6.53388950650064825619885190318, −5.17766734250840591721776738378, −4.14053057453284190747804537777, −3.03441820712828679733037239221, −1.90380721482039247499740957655,
1.90380721482039247499740957655, 3.03441820712828679733037239221, 4.14053057453284190747804537777, 5.17766734250840591721776738378, 6.53388950650064825619885190318, 7.47452512277443153862657270921, 8.148673500168146739590215078163, 9.144904734611384843436817893333, 9.814424254235537691555677314700, 10.58854000030411290247536749754