| L(s) = 1 | − 2-s − 1.56·3-s + 4-s + 2.83·5-s + 1.56·6-s + 2.44·7-s − 8-s − 0.553·9-s − 2.83·10-s + 4.72·11-s − 1.56·12-s + 2.03·13-s − 2.44·14-s − 4.44·15-s + 16-s + 5.96·17-s + 0.553·18-s − 6.97·19-s + 2.83·20-s − 3.82·21-s − 4.72·22-s − 9.19·23-s + 1.56·24-s + 3.06·25-s − 2.03·26-s + 5.55·27-s + 2.44·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.903·3-s + 0.5·4-s + 1.26·5-s + 0.638·6-s + 0.923·7-s − 0.353·8-s − 0.184·9-s − 0.897·10-s + 1.42·11-s − 0.451·12-s + 0.563·13-s − 0.652·14-s − 1.14·15-s + 0.250·16-s + 1.44·17-s + 0.130·18-s − 1.59·19-s + 0.634·20-s − 0.833·21-s − 1.00·22-s − 1.91·23-s + 0.319·24-s + 0.612·25-s − 0.398·26-s + 1.06·27-s + 0.461·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.633914786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.633914786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 - 4.72T + 11T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 - 5.96T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 + 9.19T + 23T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 37 | \( 1 - 4.14T + 37T^{2} \) |
| 41 | \( 1 - 3.62T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 3.04T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 4.32T + 67T^{2} \) |
| 71 | \( 1 + 8.16T + 71T^{2} \) |
| 73 | \( 1 + 3.30T + 73T^{2} \) |
| 79 | \( 1 + 4.15T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 5.77T + 89T^{2} \) |
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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
−8.234779295895081809259208818544, −7.38674807898936102579264094680, −6.35853115748647663319723854030, −5.99496987694517238911138290455, −5.65895601123166803627773931017, −4.53108967884250478820827688653, −3.72972222406475372343763209538, −2.31566677127295246904546678893, −1.66926916334099938508266993921, −0.832410904940511217489166207534,
0.832410904940511217489166207534, 1.66926916334099938508266993921, 2.31566677127295246904546678893, 3.72972222406475372343763209538, 4.53108967884250478820827688653, 5.65895601123166803627773931017, 5.99496987694517238911138290455, 6.35853115748647663319723854030, 7.38674807898936102579264094680, 8.234779295895081809259208818544
