| L(s) = 1 | − 1.21·3-s − 1.18·5-s − 1.93·7-s − 1.52·9-s − 11-s + 0.786·13-s + 1.43·15-s + 7.49·17-s + 19-s + 2.34·21-s − 5.07·23-s − 3.60·25-s + 5.49·27-s − 7.06·29-s − 9.73·31-s + 1.21·33-s + 2.28·35-s − 2.45·37-s − 0.954·39-s + 8.04·41-s − 8.59·43-s + 1.79·45-s − 6.36·47-s − 3.25·49-s − 9.10·51-s + 0.239·53-s + 1.18·55-s + ⋯ |
| L(s) = 1 | − 0.701·3-s − 0.527·5-s − 0.731·7-s − 0.508·9-s − 0.301·11-s + 0.218·13-s + 0.370·15-s + 1.81·17-s + 0.229·19-s + 0.512·21-s − 1.05·23-s − 0.721·25-s + 1.05·27-s − 1.31·29-s − 1.74·31-s + 0.211·33-s + 0.386·35-s − 0.404·37-s − 0.152·39-s + 1.25·41-s − 1.31·43-s + 0.268·45-s − 0.927·47-s − 0.465·49-s − 1.27·51-s + 0.0328·53-s + 0.159·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6611336523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6611336523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.21T + 3T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 7 | \( 1 + 1.93T + 7T^{2} \) |
| 13 | \( 1 - 0.786T + 13T^{2} \) |
| 17 | \( 1 - 7.49T + 17T^{2} \) |
| 23 | \( 1 + 5.07T + 23T^{2} \) |
| 29 | \( 1 + 7.06T + 29T^{2} \) |
| 31 | \( 1 + 9.73T + 31T^{2} \) |
| 37 | \( 1 + 2.45T + 37T^{2} \) |
| 41 | \( 1 - 8.04T + 41T^{2} \) |
| 43 | \( 1 + 8.59T + 43T^{2} \) |
| 47 | \( 1 + 6.36T + 47T^{2} \) |
| 53 | \( 1 - 0.239T + 53T^{2} \) |
| 59 | \( 1 - 0.984T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 1.22T + 67T^{2} \) |
| 71 | \( 1 + 3.03T + 71T^{2} \) |
| 73 | \( 1 - 7.19T + 73T^{2} \) |
| 79 | \( 1 - 3.63T + 79T^{2} \) |
| 83 | \( 1 - 9.82T + 83T^{2} \) |
| 89 | \( 1 - 0.592T + 89T^{2} \) |
| 97 | \( 1 + 4.59T + 97T^{2} \) |
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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
−8.508609085171234542910044180170, −7.79791093007114672620302951893, −7.21747196659239331490277462664, −6.14552841443891905543162858425, −5.69584703729511154327267063016, −5.01855367017662583992364139886, −3.64549001223435596327671652819, −3.39964110578570796885944592843, −1.96806810731284119319196704759, −0.48336551344959397750699491395,
0.48336551344959397750699491395, 1.96806810731284119319196704759, 3.39964110578570796885944592843, 3.64549001223435596327671652819, 5.01855367017662583992364139886, 5.69584703729511154327267063016, 6.14552841443891905543162858425, 7.21747196659239331490277462664, 7.79791093007114672620302951893, 8.508609085171234542910044180170
