| L(s) = 1 | + 2.55·2-s + 1.87·3-s + 4.50·4-s + 5-s + 4.78·6-s + 6.38·8-s + 0.514·9-s + 2.55·10-s − 11-s + 8.44·12-s + 1.50·13-s + 1.87·15-s + 7.27·16-s − 5.58·17-s + 1.31·18-s + 0.545·19-s + 4.50·20-s − 2.55·22-s + 8.58·23-s + 11.9·24-s + 25-s + 3.84·26-s − 4.66·27-s + 3.24·29-s + 4.78·30-s − 3.85·31-s + 5.78·32-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 1.08·3-s + 2.25·4-s + 0.447·5-s + 1.95·6-s + 2.25·8-s + 0.171·9-s + 0.806·10-s − 0.301·11-s + 2.43·12-s + 0.417·13-s + 0.484·15-s + 1.81·16-s − 1.35·17-s + 0.309·18-s + 0.125·19-s + 1.00·20-s − 0.543·22-s + 1.78·23-s + 2.44·24-s + 0.200·25-s + 0.753·26-s − 0.896·27-s + 0.601·29-s + 0.872·30-s − 0.692·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.401824374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.401824374\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.55T + 2T^{2} \) |
| 3 | \( 1 - 1.87T + 3T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 5.58T + 17T^{2} \) |
| 19 | \( 1 - 0.545T + 19T^{2} \) |
| 23 | \( 1 - 8.58T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 - 7.90T + 37T^{2} \) |
| 41 | \( 1 - 9.58T + 41T^{2} \) |
| 43 | \( 1 + 1.27T + 43T^{2} \) |
| 47 | \( 1 + 6.98T + 47T^{2} \) |
| 53 | \( 1 - 4.69T + 53T^{2} \) |
| 59 | \( 1 + 9.20T + 59T^{2} \) |
| 61 | \( 1 + 7.40T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 7.24T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 1.53T + 83T^{2} \) |
| 89 | \( 1 + 1.56T + 89T^{2} \) |
| 97 | \( 1 - 4.93T + 97T^{2} \) |
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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
−8.913726657479774617292759739187, −7.88139488276846107053509918605, −7.12253344560835347429421819855, −6.34330551484184339941457228512, −5.65727683135181077705797529460, −4.73544303302529411345226048576, −4.11957508799120528249307078412, −2.97781162665601635202013076140, −2.74414367324800279081540188510, −1.65790176653506851719202773251,
1.65790176653506851719202773251, 2.74414367324800279081540188510, 2.97781162665601635202013076140, 4.11957508799120528249307078412, 4.73544303302529411345226048576, 5.65727683135181077705797529460, 6.34330551484184339941457228512, 7.12253344560835347429421819855, 7.88139488276846107053509918605, 8.913726657479774617292759739187
