| L(s) = 1 | + 3-s − 1.57·7-s + 9-s + 3.88·11-s + 0.343·13-s + 6.07·17-s + 3.69·19-s − 1.57·21-s − 1.39·23-s + 27-s − 3.32·29-s − 5.25·31-s + 3.88·33-s + 8.56·37-s + 0.343·39-s − 1.27·41-s − 1.42·43-s + 0.375·47-s − 4.50·49-s + 6.07·51-s + 11.2·53-s + 3.69·57-s + 11.5·59-s − 10.9·61-s − 1.57·63-s + 10.4·67-s − 1.39·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.596·7-s + 0.333·9-s + 1.17·11-s + 0.0952·13-s + 1.47·17-s + 0.846·19-s − 0.344·21-s − 0.289·23-s + 0.192·27-s − 0.618·29-s − 0.943·31-s + 0.675·33-s + 1.40·37-s + 0.0549·39-s − 0.199·41-s − 0.217·43-s + 0.0547·47-s − 0.643·49-s + 0.850·51-s + 1.55·53-s + 0.489·57-s + 1.50·59-s − 1.40·61-s − 0.198·63-s + 1.27·67-s − 0.167·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.836118946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.836118946\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.57T + 7T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 13 | \( 1 - 0.343T + 13T^{2} \) |
| 17 | \( 1 - 6.07T + 17T^{2} \) |
| 19 | \( 1 - 3.69T + 19T^{2} \) |
| 23 | \( 1 + 1.39T + 23T^{2} \) |
| 29 | \( 1 + 3.32T + 29T^{2} \) |
| 31 | \( 1 + 5.25T + 31T^{2} \) |
| 37 | \( 1 - 8.56T + 37T^{2} \) |
| 41 | \( 1 + 1.27T + 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 - 0.375T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 4.62T + 83T^{2} \) |
| 89 | \( 1 + 7.26T + 89T^{2} \) |
| 97 | \( 1 - 6.05T + 97T^{2} \) |
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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
−7.70213825892950025805359760615, −7.39760130238245134855290762257, −6.50255044285113311146679092261, −5.87045898994292444279832379492, −5.12922450934623797522444265020, −4.01744069419983011276951595309, −3.57533533242803007425027360374, −2.85500395864954134110121963785, −1.74912414204812923917921364583, −0.864211006226909868984360382922,
0.864211006226909868984360382922, 1.74912414204812923917921364583, 2.85500395864954134110121963785, 3.57533533242803007425027360374, 4.01744069419983011276951595309, 5.12922450934623797522444265020, 5.87045898994292444279832379492, 6.50255044285113311146679092261, 7.39760130238245134855290762257, 7.70213825892950025805359760615
