| L(s) = 1 | + 2-s + 0.943·3-s + 4-s + 0.943·6-s + 4.61·7-s + 8-s − 2.10·9-s + 2.88·11-s + 0.943·12-s + 4.61·14-s + 16-s + 3.15·17-s − 2.10·18-s − 0.0635·19-s + 4.35·21-s + 2.88·22-s − 1.56·23-s + 0.943·24-s − 4.82·27-s + 4.61·28-s − 8.04·29-s − 7.30·31-s + 32-s + 2.71·33-s + 3.15·34-s − 2.10·36-s + 7.82·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.544·3-s + 0.5·4-s + 0.385·6-s + 1.74·7-s + 0.353·8-s − 0.703·9-s + 0.868·11-s + 0.272·12-s + 1.23·14-s + 0.250·16-s + 0.765·17-s − 0.497·18-s − 0.0145·19-s + 0.951·21-s + 0.614·22-s − 0.326·23-s + 0.192·24-s − 0.927·27-s + 0.872·28-s − 1.49·29-s − 1.31·31-s + 0.176·32-s + 0.473·33-s + 0.540·34-s − 0.351·36-s + 1.28·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.369624400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.369624400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.943T + 3T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 + 0.0635T + 19T^{2} \) |
| 23 | \( 1 + 1.56T + 23T^{2} \) |
| 29 | \( 1 + 8.04T + 29T^{2} \) |
| 31 | \( 1 + 7.30T + 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 - 5.61T + 47T^{2} \) |
| 53 | \( 1 + 1.26T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 4.48T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 5.03T + 71T^{2} \) |
| 73 | \( 1 - 9.09T + 73T^{2} \) |
| 79 | \( 1 - 9.16T + 79T^{2} \) |
| 83 | \( 1 - 5.55T + 83T^{2} \) |
| 89 | \( 1 + 6.75T + 89T^{2} \) |
| 97 | \( 1 - 6.91T + 97T^{2} \) |
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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
−7.80487209058199494272383896541, −7.31366446375984123650209723052, −6.20027735784817183072400295082, −5.57916435628988909661847369628, −5.08134472538735409065173229048, −4.02086891722250322222076485737, −3.79614975002246530117979947555, −2.57714082047554264632998346876, −1.97573993065906432898122718850, −1.06030989907157000096245072450,
1.06030989907157000096245072450, 1.97573993065906432898122718850, 2.57714082047554264632998346876, 3.79614975002246530117979947555, 4.02086891722250322222076485737, 5.08134472538735409065173229048, 5.57916435628988909661847369628, 6.20027735784817183072400295082, 7.31366446375984123650209723052, 7.80487209058199494272383896541
