L(s) = 1 | + 3-s − 4-s + 5-s + 9-s − 12-s + 15-s + 16-s − 2·17-s − 20-s + 25-s + 27-s − 36-s + 45-s − 2·47-s + 48-s − 2·51-s − 60-s − 64-s + 2·68-s + 75-s − 2·79-s + 80-s + 81-s − 2·83-s − 2·85-s − 100-s − 108-s + ⋯ |
L(s) = 1 | + 3-s − 4-s + 5-s + 9-s − 12-s + 15-s + 16-s − 2·17-s − 20-s + 25-s + 27-s − 36-s + 45-s − 2·47-s + 48-s − 2·51-s − 60-s − 64-s + 2·68-s + 75-s − 2·79-s + 80-s + 81-s − 2·83-s − 2·85-s − 100-s − 108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.227127003\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.227127003\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 + T )^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.20278416063923868675093066346, −9.627366540460297318138205177134, −8.842548470518722242354235233287, −8.419387340681146725262179475931, −7.16221285282161101246575092961, −6.21189580242390673333307389569, −4.96090470011521248126898849684, −4.20969157815888785241273240518, −2.94217195339892845878731375317, −1.75562792641656088192317090538,
1.75562792641656088192317090538, 2.94217195339892845878731375317, 4.20969157815888785241273240518, 4.96090470011521248126898849684, 6.21189580242390673333307389569, 7.16221285282161101246575092961, 8.419387340681146725262179475931, 8.842548470518722242354235233287, 9.627366540460297318138205177134, 10.20278416063923868675093066346