L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s − 14-s + 16-s − 2·17-s + 19-s + 20-s − 22-s − 23-s − 28-s + 31-s + 32-s − 2·34-s − 35-s − 37-s + 38-s + 40-s + 41-s − 44-s − 46-s + 49-s − 55-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s − 14-s + 16-s − 2·17-s + 19-s + 20-s − 22-s − 23-s − 28-s + 31-s + 32-s − 2·34-s − 35-s − 37-s + 38-s + 40-s + 41-s − 44-s − 46-s + 49-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.753521755\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.753521755\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 + T )^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−10.48860615324319911172915212623, −9.957349023874567555417663990553, −8.984313395998733290831877798221, −7.72215874108065267747623198746, −6.71490057619476191556249838639, −6.07463380538032471166318897736, −5.26800699997401246927775108992, −4.19440565051837915297271177807, −2.91867617249410807579486425175, −2.11132364390146111360862922278,
2.11132364390146111360862922278, 2.91867617249410807579486425175, 4.19440565051837915297271177807, 5.26800699997401246927775108992, 6.07463380538032471166318897736, 6.71490057619476191556249838639, 7.72215874108065267747623198746, 8.984313395998733290831877798221, 9.957349023874567555417663990553, 10.48860615324319911172915212623