L(s) = 1 | + 2.05·3-s − 5-s + 4.29·7-s + 1.22·9-s + 3.31·11-s + 5.16·13-s − 2.05·15-s + 4.00·17-s + 1.43·19-s + 8.82·21-s + 3.52·23-s + 25-s − 3.65·27-s + 4.87·29-s + 3.50·31-s + 6.81·33-s − 4.29·35-s + 2.72·37-s + 10.6·39-s + 6.30·41-s − 11.1·43-s − 1.22·45-s − 4.78·47-s + 11.4·49-s + 8.21·51-s − 12.3·53-s − 3.31·55-s + ⋯ |
L(s) = 1 | + 1.18·3-s − 0.447·5-s + 1.62·7-s + 0.406·9-s + 1.00·11-s + 1.43·13-s − 0.530·15-s + 0.970·17-s + 0.328·19-s + 1.92·21-s + 0.735·23-s + 0.200·25-s − 0.703·27-s + 0.905·29-s + 0.628·31-s + 1.18·33-s − 0.726·35-s + 0.448·37-s + 1.70·39-s + 0.985·41-s − 1.70·43-s − 0.181·45-s − 0.698·47-s + 1.63·49-s + 1.15·51-s − 1.68·53-s − 0.447·55-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
\(L(1)\) |
\(\approx\) |
\(4.627364852\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.627364852\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.05T + 3T^{2} \) |
| 7 | \( 1 - 4.29T + 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 13 | \( 1 - 5.16T + 13T^{2} \) |
| 17 | \( 1 - 4.00T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 - 3.52T + 23T^{2} \) |
| 29 | \( 1 - 4.87T + 29T^{2} \) |
| 31 | \( 1 - 3.50T + 31T^{2} \) |
| 37 | \( 1 - 2.72T + 37T^{2} \) |
| 41 | \( 1 - 6.30T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 4.78T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 1.84T + 59T^{2} \) |
| 61 | \( 1 + 2.32T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 0.334T + 73T^{2} \) |
| 79 | \( 1 + 6.03T + 79T^{2} \) |
| 83 | \( 1 + 2.10T + 83T^{2} \) |
| 89 | \( 1 - 4.28T + 89T^{2} \) |
| 97 | \( 1 - 6.01T + 97T^{2} \) |
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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
−8.050499827723473249610574656552, −7.45241181579798677179033787632, −6.53788856766547605143804746487, −5.75408404672243825485265326789, −4.80233312557753810799005786832, −4.21905066846257358671483971338, −3.41561631996508612257242918581, −2.86353114173836110635368558014, −1.55035862371756954590637367411, −1.22564552990132591845438335106,
1.22564552990132591845438335106, 1.55035862371756954590637367411, 2.86353114173836110635368558014, 3.41561631996508612257242918581, 4.21905066846257358671483971338, 4.80233312557753810799005786832, 5.75408404672243825485265326789, 6.53788856766547605143804746487, 7.45241181579798677179033787632, 8.050499827723473249610574656552