L(s) = 1 | − 0.505·3-s − 5-s − 0.615·7-s − 2.74·9-s − 0.537·11-s − 6.49·13-s + 0.505·15-s − 4.51·17-s + 4.17·19-s + 0.311·21-s − 2.96·23-s + 25-s + 2.90·27-s − 6.55·29-s + 7.16·31-s + 0.271·33-s + 0.615·35-s − 7.23·37-s + 3.28·39-s − 3.15·41-s − 8.42·43-s + 2.74·45-s + 0.179·47-s − 6.62·49-s + 2.28·51-s − 8.16·53-s + 0.537·55-s + ⋯ |
L(s) = 1 | − 0.291·3-s − 0.447·5-s − 0.232·7-s − 0.914·9-s − 0.161·11-s − 1.80·13-s + 0.130·15-s − 1.09·17-s + 0.958·19-s + 0.0678·21-s − 0.617·23-s + 0.200·25-s + 0.558·27-s − 1.21·29-s + 1.28·31-s + 0.0472·33-s + 0.103·35-s − 1.19·37-s + 0.526·39-s − 0.492·41-s − 1.28·43-s + 0.409·45-s + 0.0262·47-s − 0.945·49-s + 0.319·51-s − 1.12·53-s + 0.0724·55-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
\(L(1)\) |
\(\approx\) |
\(0.3631405427\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3631405427\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.505T + 3T^{2} \) |
| 7 | \( 1 + 0.615T + 7T^{2} \) |
| 11 | \( 1 + 0.537T + 11T^{2} \) |
| 13 | \( 1 + 6.49T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 + 6.55T + 29T^{2} \) |
| 31 | \( 1 - 7.16T + 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 + 3.15T + 41T^{2} \) |
| 43 | \( 1 + 8.42T + 43T^{2} \) |
| 47 | \( 1 - 0.179T + 47T^{2} \) |
| 53 | \( 1 + 8.16T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 + 6.12T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 6.93T + 79T^{2} \) |
| 83 | \( 1 + 1.01T + 83T^{2} \) |
| 89 | \( 1 - 6.88T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.87812116576129185932289238348, −7.03107384462119255696822021045, −6.62371222208512167855924734051, −5.60916968197580627875028113483, −5.07391076427207027190340204150, −4.43819114921499842302320416582, −3.39382012493274109714449031089, −2.74376479511500961037889480165, −1.87119029351690110542598439400, −0.28311471137300951153611510176,
0.28311471137300951153611510176, 1.87119029351690110542598439400, 2.74376479511500961037889480165, 3.39382012493274109714449031089, 4.43819114921499842302320416582, 5.07391076427207027190340204150, 5.60916968197580627875028113483, 6.62371222208512167855924734051, 7.03107384462119255696822021045, 7.87812116576129185932289238348