| L(s) = 1 | + 3-s + 4.62·7-s + 9-s − 4.94·11-s − 3.76·13-s − 2.69·17-s + 5.87·19-s + 4.62·21-s + 6.67·23-s + 27-s + 1.20·29-s + 3.30·31-s − 4.94·33-s + 1.87·37-s − 3.76·39-s − 3.03·41-s − 10.6·43-s + 0.259·47-s + 14.4·49-s − 2.69·51-s + 9.79·53-s + 5.87·57-s − 9.62·59-s + 6.27·61-s + 4.62·63-s + 2.56·67-s + 6.67·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.74·7-s + 0.333·9-s − 1.49·11-s − 1.04·13-s − 0.652·17-s + 1.34·19-s + 1.00·21-s + 1.39·23-s + 0.192·27-s + 0.222·29-s + 0.593·31-s − 0.861·33-s + 0.308·37-s − 0.602·39-s − 0.473·41-s − 1.62·43-s + 0.0378·47-s + 2.05·49-s − 0.376·51-s + 1.34·53-s + 0.777·57-s − 1.25·59-s + 0.803·61-s + 0.582·63-s + 0.312·67-s + 0.803·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.955584822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.955584822\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.62T + 7T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 + 3.76T + 13T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 - 6.67T + 23T^{2} \) |
| 29 | \( 1 - 1.20T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 37 | \( 1 - 1.87T + 37T^{2} \) |
| 41 | \( 1 + 3.03T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 0.259T + 47T^{2} \) |
| 53 | \( 1 - 9.79T + 53T^{2} \) |
| 59 | \( 1 + 9.62T + 59T^{2} \) |
| 61 | \( 1 - 6.27T + 61T^{2} \) |
| 67 | \( 1 - 2.56T + 67T^{2} \) |
| 71 | \( 1 - 8.67T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 8.89T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 3.98T + 97T^{2} \) |
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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
−7.84496411776893161324584138064, −7.45783575979243570363030303199, −6.75839812709099576082076634404, −5.42971893528120788697207710089, −4.95903008070247618877328583082, −4.65645983917090192668180925279, −3.38704513784868412685089814794, −2.57684528509800867980893571236, −1.97076871985965162129799159134, −0.849336603561742906430404392882,
0.849336603561742906430404392882, 1.97076871985965162129799159134, 2.57684528509800867980893571236, 3.38704513784868412685089814794, 4.65645983917090192668180925279, 4.95903008070247618877328583082, 5.42971893528120788697207710089, 6.75839812709099576082076634404, 7.45783575979243570363030303199, 7.84496411776893161324584138064
