L(s) = 1 | + 1.91·3-s + 3.63·5-s − 4.69·7-s + 0.651·9-s + 3.01·11-s + 5.45·13-s + 6.94·15-s + 17-s − 2.51·19-s − 8.96·21-s − 2.25·23-s + 8.20·25-s − 4.48·27-s + 9.37·29-s + 6.90·31-s + 5.75·33-s − 17.0·35-s + 0.180·37-s + 10.4·39-s − 0.746·41-s − 6.45·43-s + 2.36·45-s − 11.6·47-s + 15.0·49-s + 1.91·51-s − 1.10·53-s + 10.9·55-s + ⋯ |
L(s) = 1 | + 1.10·3-s + 1.62·5-s − 1.77·7-s + 0.217·9-s + 0.907·11-s + 1.51·13-s + 1.79·15-s + 0.242·17-s − 0.576·19-s − 1.95·21-s − 0.471·23-s + 1.64·25-s − 0.863·27-s + 1.74·29-s + 1.23·31-s + 1.00·33-s − 2.88·35-s + 0.0297·37-s + 1.66·39-s − 0.116·41-s − 0.983·43-s + 0.353·45-s − 1.69·47-s + 2.14·49-s + 0.267·51-s − 0.151·53-s + 1.47·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.015063959\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.015063959\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 1.91T + 3T^{2} \) |
| 5 | \( 1 - 3.63T + 5T^{2} \) |
| 7 | \( 1 + 4.69T + 7T^{2} \) |
| 11 | \( 1 - 3.01T + 11T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 31 | \( 1 - 6.90T + 31T^{2} \) |
| 37 | \( 1 - 0.180T + 37T^{2} \) |
| 41 | \( 1 + 0.746T + 41T^{2} \) |
| 43 | \( 1 + 6.45T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 1.10T + 53T^{2} \) |
| 61 | \( 1 - 0.347T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 9.72T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 7.14T + 79T^{2} \) |
| 83 | \( 1 - 1.12T + 83T^{2} \) |
| 89 | \( 1 - 2.80T + 89T^{2} \) |
| 97 | \( 1 + 4.73T + 97T^{2} \) |
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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
−8.290587857262538835707299132405, −6.70212999680540107113554443062, −6.44953852522068756487027372326, −6.13964040462893190563585611301, −5.14708011341206729058480051936, −3.92988657537777471317571223015, −3.36526525853061322293121343659, −2.73494377409663472211544081923, −1.95427672117851446141375614908, −0.968055336734428193033877081776,
0.968055336734428193033877081776, 1.95427672117851446141375614908, 2.73494377409663472211544081923, 3.36526525853061322293121343659, 3.92988657537777471317571223015, 5.14708011341206729058480051936, 6.13964040462893190563585611301, 6.44953852522068756487027372326, 6.70212999680540107113554443062, 8.290587857262538835707299132405