L(s) = 1 | − 3-s − 3.46·5-s − 3.46·7-s + 9-s + 3.46·15-s + 6·17-s − 4·19-s + 3.46·21-s + 6.92·23-s + 6.99·25-s − 27-s + 3.46·29-s − 3.46·31-s + 11.9·35-s + 6.92·37-s + 6·41-s − 4·43-s − 3.46·45-s − 6.92·47-s + 4.99·49-s − 6·51-s + 3.46·53-s + 4·57-s + 12·59-s − 6.92·61-s − 3.46·63-s + 4·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.54·5-s − 1.30·7-s + 0.333·9-s + 0.894·15-s + 1.45·17-s − 0.917·19-s + 0.755·21-s + 1.44·23-s + 1.39·25-s − 0.192·27-s + 0.643·29-s − 0.622·31-s + 2.02·35-s + 1.13·37-s + 0.937·41-s − 0.609·43-s − 0.516·45-s − 1.01·47-s + 0.714·49-s − 0.840·51-s + 0.475·53-s + 0.529·57-s + 1.56·59-s − 0.887·61-s − 0.436·63-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6700896478\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6700896478\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.40832323178950496516690073363, −9.567824670349199995278608367568, −8.546702118283194580045683006223, −7.60853996407630508849648715281, −6.89673040000006302638164559861, −6.01226049225962676935830219852, −4.82532112152912354925956242158, −3.79580701790612401048742900272, −3.02866888416199003373007631606, −0.68078562470568089929410367530,
0.68078562470568089929410367530, 3.02866888416199003373007631606, 3.79580701790612401048742900272, 4.82532112152912354925956242158, 6.01226049225962676935830219852, 6.89673040000006302638164559861, 7.60853996407630508849648715281, 8.546702118283194580045683006223, 9.567824670349199995278608367568, 10.40832323178950496516690073363