L(s) = 1 | + 3-s + 9-s − 13-s − 19-s + 25-s + 27-s − 31-s − 37-s − 39-s − 43-s − 57-s + 2·61-s − 67-s − 73-s + 75-s − 79-s + 81-s − 93-s + 2·97-s − 103-s − 109-s − 111-s − 117-s + ⋯ |
L(s) = 1 | + 3-s + 9-s − 13-s − 19-s + 25-s + 27-s − 31-s − 37-s − 39-s − 43-s − 57-s + 2·61-s − 67-s − 73-s + 75-s − 79-s + 81-s − 93-s + 2·97-s − 103-s − 109-s − 111-s − 117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.203191104\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.203191104\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.66179444964847131016268260418, −9.992304814917284726116447913179, −9.041261215796509979421627909881, −8.400122770178731149198343958187, −7.37835468379044894035559116671, −6.68546487276271598199087656551, −5.21020278778570492589089292909, −4.20249201282363993133386632534, −3.05186481588102742617886479608, −1.93605987703093487834817760007,
1.93605987703093487834817760007, 3.05186481588102742617886479608, 4.20249201282363993133386632534, 5.21020278778570492589089292909, 6.68546487276271598199087656551, 7.37835468379044894035559116671, 8.400122770178731149198343958187, 9.041261215796509979421627909881, 9.992304814917284726116447913179, 10.66179444964847131016268260418