| L(s) = 1 | + 3-s − 2.75·5-s − 0.925·7-s + 9-s − 1.99·11-s − 4.40·13-s − 2.75·15-s − 4.21·17-s − 8.42·19-s − 0.925·21-s − 7.31·23-s + 2.61·25-s + 27-s + 4.78·29-s + 4.08·31-s − 1.99·33-s + 2.55·35-s − 1.49·37-s − 4.40·39-s + 7.65·41-s + 10.8·43-s − 2.75·45-s − 1.93·47-s − 6.14·49-s − 4.21·51-s + 11.4·53-s + 5.50·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.23·5-s − 0.349·7-s + 0.333·9-s − 0.601·11-s − 1.22·13-s − 0.712·15-s − 1.02·17-s − 1.93·19-s − 0.201·21-s − 1.52·23-s + 0.522·25-s + 0.192·27-s + 0.888·29-s + 0.733·31-s − 0.347·33-s + 0.431·35-s − 0.246·37-s − 0.705·39-s + 1.19·41-s + 1.65·43-s − 0.411·45-s − 0.282·47-s − 0.877·49-s − 0.590·51-s + 1.57·53-s + 0.742·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7636344951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7636344951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 + 2.75T + 5T^{2} \) |
| 7 | \( 1 + 0.925T + 7T^{2} \) |
| 11 | \( 1 + 1.99T + 11T^{2} \) |
| 13 | \( 1 + 4.40T + 13T^{2} \) |
| 17 | \( 1 + 4.21T + 17T^{2} \) |
| 19 | \( 1 + 8.42T + 19T^{2} \) |
| 23 | \( 1 + 7.31T + 23T^{2} \) |
| 29 | \( 1 - 4.78T + 29T^{2} \) |
| 31 | \( 1 - 4.08T + 31T^{2} \) |
| 37 | \( 1 + 1.49T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 1.93T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 3.05T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 + 8.05T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 0.478T + 83T^{2} \) |
| 89 | \( 1 - 8.77T + 89T^{2} \) |
| 97 | \( 1 - 1.80T + 97T^{2} \) |
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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
−8.016743695238107039551259120428, −7.58578472425649582259247244876, −6.74765576246488847029425431763, −6.14911243876590368437548817211, −4.92694965743446204710502618945, −4.24041800172072944104772726433, −3.85605191670719296772541830150, −2.57475059060590744196210594406, −2.26830615802004826803143709174, −0.41246691211841480147740581512,
0.41246691211841480147740581512, 2.26830615802004826803143709174, 2.57475059060590744196210594406, 3.85605191670719296772541830150, 4.24041800172072944104772726433, 4.92694965743446204710502618945, 6.14911243876590368437548817211, 6.74765576246488847029425431763, 7.58578472425649582259247244876, 8.016743695238107039551259120428
