| L(s) = 1 | − 3.19·3-s + 1.68·5-s + 2.62·7-s + 7.20·9-s − 5.62·11-s − 3.12·13-s − 5.39·15-s − 2.89·17-s − 4.77·19-s − 8.39·21-s + 1.59·23-s − 2.14·25-s − 13.4·27-s − 1.80·29-s − 10.0·31-s + 17.9·33-s + 4.44·35-s − 5.63·37-s + 9.98·39-s + 7.93·41-s + 7.23·43-s + 12.1·45-s + 47-s − 0.0915·49-s + 9.23·51-s + 10.6·53-s − 9.50·55-s + ⋯ |
| L(s) = 1 | − 1.84·3-s + 0.755·5-s + 0.993·7-s + 2.40·9-s − 1.69·11-s − 0.866·13-s − 1.39·15-s − 0.701·17-s − 1.09·19-s − 1.83·21-s + 0.333·23-s − 0.429·25-s − 2.58·27-s − 0.335·29-s − 1.79·31-s + 3.12·33-s + 0.750·35-s − 0.926·37-s + 1.59·39-s + 1.23·41-s + 1.10·43-s + 1.81·45-s + 0.145·47-s − 0.0130·49-s + 1.29·51-s + 1.46·53-s − 1.28·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6571392909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6571392909\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 3 | \( 1 + 3.19T + 3T^{2} \) |
| 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 11 | \( 1 + 5.62T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 + 4.77T + 19T^{2} \) |
| 23 | \( 1 - 1.59T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 5.63T + 37T^{2} \) |
| 41 | \( 1 - 7.93T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 6.67T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 8.03T + 73T^{2} \) |
| 79 | \( 1 - 3.93T + 79T^{2} \) |
| 83 | \( 1 - 9.52T + 83T^{2} \) |
| 89 | \( 1 + 2.12T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−7.74587783678518269951607449550, −7.33881993185465019638664952151, −6.51976423881955055134300710851, −5.66152640235367102137681806516, −5.36267852764423440986476456171, −4.78984236654249988411145928373, −4.05666184919272512715214334495, −2.34440399044740361254549855053, −1.85345385588301704613971592407, −0.45213861557637625984334934791,
0.45213861557637625984334934791, 1.85345385588301704613971592407, 2.34440399044740361254549855053, 4.05666184919272512715214334495, 4.78984236654249988411145928373, 5.36267852764423440986476456171, 5.66152640235367102137681806516, 6.51976423881955055134300710851, 7.33881993185465019638664952151, 7.74587783678518269951607449550
