L(s) = 1 | + 0.732·3-s − 5-s + 3.46·7-s − 2.46·9-s − 3.46·11-s − 3.26·13-s − 0.732·15-s − 0.732·17-s − 6.92·19-s + 2.53·21-s − 2.19·23-s + 25-s − 4·27-s − 1.46·29-s + 1.46·31-s − 2.53·33-s − 3.46·35-s + 10.1·37-s − 2.39·39-s + 8.39·41-s − 2·43-s + 2.46·45-s + 4.92·47-s + 4.99·49-s − 0.535·51-s + 1.80·53-s + 3.46·55-s + ⋯ |
L(s) = 1 | + 0.422·3-s − 0.447·5-s + 1.30·7-s − 0.821·9-s − 1.04·11-s − 0.906·13-s − 0.189·15-s − 0.177·17-s − 1.58·19-s + 0.553·21-s − 0.457·23-s + 0.200·25-s − 0.769·27-s − 0.271·29-s + 0.262·31-s − 0.441·33-s − 0.585·35-s + 1.67·37-s − 0.383·39-s + 1.31·41-s − 0.304·43-s + 0.367·45-s + 0.718·47-s + 0.714·49-s − 0.0750·51-s + 0.247·53-s + 0.467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.566746697\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.566746697\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 3.26T + 13T^{2} \) |
| 17 | \( 1 + 0.732T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 4.92T + 47T^{2} \) |
| 53 | \( 1 - 1.80T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 2.19T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 7.26T + 97T^{2} \) |
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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
−7.74733845697301178225989411630, −7.63079997405439544254797238383, −6.39004813766133549467414595153, −5.72236520779629232424433432274, −4.81313050260009395957652222512, −4.49391564219667765674273899465, −3.48958739474804248679747300250, −2.33157528550292942006398605068, −2.22968582082182138094575734697, −0.57271194669710679835503882252,
0.57271194669710679835503882252, 2.22968582082182138094575734697, 2.33157528550292942006398605068, 3.48958739474804248679747300250, 4.49391564219667765674273899465, 4.81313050260009395957652222512, 5.72236520779629232424433432274, 6.39004813766133549467414595153, 7.63079997405439544254797238383, 7.74733845697301178225989411630