| L(s) = 1 | + 5.23·7-s − 7.70·23-s + 6·29-s − 4.47·41-s + 6.76·43-s − 0.291·47-s + 20.4·49-s + 13.4·61-s + 14.1·67-s + 4.29·83-s − 6·89-s + 18·101-s − 2.18·103-s + 19.7·107-s − 13.4·109-s + ⋯ |
| L(s) = 1 | + 1.97·7-s − 1.60·23-s + 1.11·29-s − 0.698·41-s + 1.03·43-s − 0.0425·47-s + 2.91·49-s + 1.71·61-s + 1.73·67-s + 0.471·83-s − 0.635·89-s + 1.79·101-s − 0.214·103-s + 1.90·107-s − 1.28·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.706710580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.706710580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5.23T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 6.76T + 43T^{2} \) |
| 47 | \( 1 + 0.291T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4.29T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.096559439968840208930334390025, −7.38196287531328426459859164677, −6.55304506873024720074428570896, −5.67834446320787553307869061323, −5.07503917300434549897758671992, −4.40062423072192247189511174106, −3.76033628270095452825522072513, −2.46129744287166072400837008542, −1.84434307512047331684423106027, −0.869168304842140774857568928108,
0.869168304842140774857568928108, 1.84434307512047331684423106027, 2.46129744287166072400837008542, 3.76033628270095452825522072513, 4.40062423072192247189511174106, 5.07503917300434549897758671992, 5.67834446320787553307869061323, 6.55304506873024720074428570896, 7.38196287531328426459859164677, 8.096559439968840208930334390025
