L(s) = 1 | − 2-s − 5-s + 8-s + 10-s + 2·11-s + 13-s − 16-s − 2·22-s − 26-s − 40-s + 2·41-s − 43-s − 47-s + 49-s − 2·55-s − 59-s − 61-s + 64-s − 65-s − 71-s + 2·79-s + 80-s − 2·82-s − 83-s + 86-s + 2·88-s − 89-s + ⋯ |
L(s) = 1 | − 2-s − 5-s + 8-s + 10-s + 2·11-s + 13-s − 16-s − 2·22-s − 26-s − 40-s + 2·41-s − 43-s − 47-s + 49-s − 2·55-s − 59-s − 61-s + 64-s − 65-s − 71-s + 2·79-s + 80-s − 2·82-s − 83-s + 86-s + 2·88-s − 89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4462802668\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4462802668\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 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 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 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
−11.48120080661411090532373695491, −10.83338377562048097070994610480, −9.588945031591092329971096479302, −8.931287185736333318375210294355, −8.160802915082779291609672451907, −7.22765273448491413273119793205, −6.19075407689203210695833474270, −4.41407558466250751498047916509, −3.68766173148466057722643549840, −1.31924089132326928898750164297,
1.31924089132326928898750164297, 3.68766173148466057722643549840, 4.41407558466250751498047916509, 6.19075407689203210695833474270, 7.22765273448491413273119793205, 8.160802915082779291609672451907, 8.931287185736333318375210294355, 9.588945031591092329971096479302, 10.83338377562048097070994610480, 11.48120080661411090532373695491