| L(s) = 1 | + 1.06·3-s + 1.33·5-s − 4.58·7-s − 1.86·9-s − 0.0331·11-s + 0.479·13-s + 1.42·15-s − 5.91·17-s − 4.39·19-s − 4.89·21-s + 7.03·23-s − 3.21·25-s − 5.18·27-s + 0.343·29-s + 3.28·31-s − 0.0353·33-s − 6.13·35-s − 0.698·37-s + 0.511·39-s + 11.5·41-s − 0.748·43-s − 2.49·45-s + 8.97·47-s + 14.0·49-s − 6.30·51-s + 3.83·53-s − 0.0443·55-s + ⋯ |
| L(s) = 1 | + 0.615·3-s + 0.598·5-s − 1.73·7-s − 0.621·9-s − 0.0100·11-s + 0.132·13-s + 0.368·15-s − 1.43·17-s − 1.00·19-s − 1.06·21-s + 1.46·23-s − 0.642·25-s − 0.997·27-s + 0.0638·29-s + 0.589·31-s − 0.00616·33-s − 1.03·35-s − 0.114·37-s + 0.0818·39-s + 1.80·41-s − 0.114·43-s − 0.371·45-s + 1.30·47-s + 2.00·49-s − 0.883·51-s + 0.527·53-s − 0.00598·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.589595314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.589595314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 1.06T + 3T^{2} \) |
| 5 | \( 1 - 1.33T + 5T^{2} \) |
| 7 | \( 1 + 4.58T + 7T^{2} \) |
| 11 | \( 1 + 0.0331T + 11T^{2} \) |
| 13 | \( 1 - 0.479T + 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 + 4.39T + 19T^{2} \) |
| 23 | \( 1 - 7.03T + 23T^{2} \) |
| 29 | \( 1 - 0.343T + 29T^{2} \) |
| 31 | \( 1 - 3.28T + 31T^{2} \) |
| 37 | \( 1 + 0.698T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 0.748T + 43T^{2} \) |
| 47 | \( 1 - 8.97T + 47T^{2} \) |
| 53 | \( 1 - 3.83T + 53T^{2} \) |
| 59 | \( 1 - 5.84T + 59T^{2} \) |
| 61 | \( 1 - 3.93T + 61T^{2} \) |
| 67 | \( 1 - 4.58T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 1.25T + 73T^{2} \) |
| 79 | \( 1 - 7.66T + 79T^{2} \) |
| 83 | \( 1 - 8.00T + 83T^{2} \) |
| 89 | \( 1 + 2.97T + 89T^{2} \) |
| 97 | \( 1 - 0.953T + 97T^{2} \) |
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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
−8.259928757223650630575509747516, −7.24233703556918079321652040185, −6.55720985764032952014223810704, −6.13549901893578084008829482693, −5.36539169714644306288736805626, −4.22745254747647023893063138851, −3.55763372551641838992058490414, −2.58642404890428431157261819741, −2.31829089060837174776246683206, −0.60195657217841849445873436658,
0.60195657217841849445873436658, 2.31829089060837174776246683206, 2.58642404890428431157261819741, 3.55763372551641838992058490414, 4.22745254747647023893063138851, 5.36539169714644306288736805626, 6.13549901893578084008829482693, 6.55720985764032952014223810704, 7.24233703556918079321652040185, 8.259928757223650630575509747516
