| L(s) = 1 | + 2.40·2-s + 3.78·4-s − 2.94·5-s + 2.89·7-s + 4.28·8-s − 7.08·10-s + 2.53·11-s + 5.47·13-s + 6.95·14-s + 2.72·16-s + 3.33·17-s − 5.49·19-s − 11.1·20-s + 6.09·22-s − 23-s + 3.68·25-s + 13.1·26-s + 10.9·28-s − 29-s + 5.60·31-s − 1.99·32-s + 8.02·34-s − 8.52·35-s + 4.06·37-s − 13.2·38-s − 12.6·40-s + 9.72·41-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.89·4-s − 1.31·5-s + 1.09·7-s + 1.51·8-s − 2.23·10-s + 0.764·11-s + 1.51·13-s + 1.85·14-s + 0.682·16-s + 0.809·17-s − 1.25·19-s − 2.49·20-s + 1.29·22-s − 0.208·23-s + 0.736·25-s + 2.58·26-s + 2.06·28-s − 0.185·29-s + 1.00·31-s − 0.353·32-s + 1.37·34-s − 1.44·35-s + 0.668·37-s − 2.14·38-s − 1.99·40-s + 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.848915027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.848915027\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 5 | \( 1 + 2.94T + 5T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 11 | \( 1 - 2.53T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 + 5.49T + 19T^{2} \) |
| 31 | \( 1 - 5.60T + 31T^{2} \) |
| 37 | \( 1 - 4.06T + 37T^{2} \) |
| 41 | \( 1 - 9.72T + 41T^{2} \) |
| 43 | \( 1 - 2.73T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 1.45T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 5.14T + 67T^{2} \) |
| 71 | \( 1 - 3.43T + 71T^{2} \) |
| 73 | \( 1 - 6.90T + 73T^{2} \) |
| 79 | \( 1 + 5.51T + 79T^{2} \) |
| 83 | \( 1 + 0.148T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
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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
−8.096535799075881310961659541597, −7.23058230224886783874242936250, −6.34183450451696268457192564690, −5.97016075762497640550430004087, −4.90484997414720531346627231424, −4.34648614955497208287769134967, −3.85388251124691297737006134477, −3.26934830013435512361120765501, −2.10375909820367311019808793449, −1.05335745834331955540234556291,
1.05335745834331955540234556291, 2.10375909820367311019808793449, 3.26934830013435512361120765501, 3.85388251124691297737006134477, 4.34648614955497208287769134967, 4.90484997414720531346627231424, 5.97016075762497640550430004087, 6.34183450451696268457192564690, 7.23058230224886783874242936250, 8.096535799075881310961659541597
