| L(s) = 1 | − 2-s − 3.41·3-s + 4-s − 1.05·5-s + 3.41·6-s − 0.874·7-s − 8-s + 8.69·9-s + 1.05·10-s + 1.91·11-s − 3.41·12-s − 4.30·13-s + 0.874·14-s + 3.62·15-s + 16-s − 2.56·17-s − 8.69·18-s − 2.73·19-s − 1.05·20-s + 2.99·21-s − 1.91·22-s − 23-s + 3.41·24-s − 3.87·25-s + 4.30·26-s − 19.4·27-s − 0.874·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.97·3-s + 0.5·4-s − 0.474·5-s + 1.39·6-s − 0.330·7-s − 0.353·8-s + 2.89·9-s + 0.335·10-s + 0.578·11-s − 0.987·12-s − 1.19·13-s + 0.233·14-s + 0.935·15-s + 0.250·16-s − 0.621·17-s − 2.04·18-s − 0.628·19-s − 0.237·20-s + 0.652·21-s − 0.409·22-s − 0.208·23-s + 0.698·24-s − 0.775·25-s + 0.845·26-s − 3.74·27-s − 0.165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2523669336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2523669336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 3.41T + 3T^{2} \) |
| 5 | \( 1 + 1.05T + 5T^{2} \) |
| 7 | \( 1 + 0.874T + 7T^{2} \) |
| 11 | \( 1 - 1.91T + 11T^{2} \) |
| 13 | \( 1 + 4.30T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 - 0.420T + 37T^{2} \) |
| 41 | \( 1 + 6.84T + 41T^{2} \) |
| 43 | \( 1 - 6.86T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 2.54T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 1.04T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 + 5.69T + 83T^{2} \) |
| 89 | \( 1 + 0.816T + 89T^{2} \) |
| 97 | \( 1 + 9.44T + 97T^{2} \) |
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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
−7.889010143810902662210112155999, −7.18072594441833539374850876374, −6.61697539981076485045810673053, −6.18631326424925315403810930329, −5.30753304214824248824843456561, −4.52553766434621895385941465015, −3.98824182220043400272468045337, −2.52292367476401224787173664168, −1.39274631343707675968806916359, −0.34673927334959595109901155848,
0.34673927334959595109901155848, 1.39274631343707675968806916359, 2.52292367476401224787173664168, 3.98824182220043400272468045337, 4.52553766434621895385941465015, 5.30753304214824248824843456561, 6.18631326424925315403810930329, 6.61697539981076485045810673053, 7.18072594441833539374850876374, 7.889010143810902662210112155999
