| L(s) = 1 | − 2.69·3-s + 1.28·5-s − 4.83·7-s + 4.27·9-s + 2.78·11-s + 0.338·13-s − 3.46·15-s − 0.869·17-s − 4.65·19-s + 13.0·21-s + 1.04·23-s − 3.35·25-s − 3.45·27-s + 4.54·29-s + 8.48·31-s − 7.51·33-s − 6.20·35-s − 8.31·37-s − 0.912·39-s + 1.30·41-s − 5.41·43-s + 5.49·45-s − 3.76·47-s + 16.3·49-s + 2.34·51-s − 5.78·53-s + 3.57·55-s + ⋯ |
| L(s) = 1 | − 1.55·3-s + 0.573·5-s − 1.82·7-s + 1.42·9-s + 0.840·11-s + 0.0937·13-s − 0.894·15-s − 0.210·17-s − 1.06·19-s + 2.84·21-s + 0.218·23-s − 0.670·25-s − 0.664·27-s + 0.843·29-s + 1.52·31-s − 1.30·33-s − 1.04·35-s − 1.36·37-s − 0.146·39-s + 0.203·41-s − 0.826·43-s + 0.818·45-s − 0.548·47-s + 2.34·49-s + 0.328·51-s − 0.795·53-s + 0.482·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 + 4.83T + 7T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 13 | \( 1 - 0.338T + 13T^{2} \) |
| 17 | \( 1 + 0.869T + 17T^{2} \) |
| 19 | \( 1 + 4.65T + 19T^{2} \) |
| 23 | \( 1 - 1.04T + 23T^{2} \) |
| 29 | \( 1 - 4.54T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 + 3.76T + 47T^{2} \) |
| 53 | \( 1 + 5.78T + 53T^{2} \) |
| 59 | \( 1 - 7.14T + 59T^{2} \) |
| 61 | \( 1 + 0.121T + 61T^{2} \) |
| 67 | \( 1 - 3.18T + 67T^{2} \) |
| 71 | \( 1 - 8.89T + 71T^{2} \) |
| 73 | \( 1 - 8.55T + 73T^{2} \) |
| 79 | \( 1 - 6.42T + 79T^{2} \) |
| 83 | \( 1 - 6.54T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 6.17T + 97T^{2} \) |
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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
−6.88021141463571128762411337968, −6.58944921334227509820096157857, −6.28997101962520079730770990663, −5.60169727746613693447632096066, −4.80155696508797170219956271312, −3.99869348729991018663179888501, −3.20237485633936196903381503054, −2.12184467178330405559880656232, −0.930683340186500262459462737525, 0,
0.930683340186500262459462737525, 2.12184467178330405559880656232, 3.20237485633936196903381503054, 3.99869348729991018663179888501, 4.80155696508797170219956271312, 5.60169727746613693447632096066, 6.28997101962520079730770990663, 6.58944921334227509820096157857, 6.88021141463571128762411337968
