| L(s) = 1 | − 2-s − 3-s + 4-s + 3.45·5-s + 6-s + 3.88·7-s − 8-s + 9-s − 3.45·10-s − 5.60·11-s − 12-s − 4.12·13-s − 3.88·14-s − 3.45·15-s + 16-s + 17-s − 18-s − 5.92·19-s + 3.45·20-s − 3.88·21-s + 5.60·22-s − 3.68·23-s + 24-s + 6.95·25-s + 4.12·26-s − 27-s + 3.88·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.54·5-s + 0.408·6-s + 1.46·7-s − 0.353·8-s + 0.333·9-s − 1.09·10-s − 1.68·11-s − 0.288·12-s − 1.14·13-s − 1.03·14-s − 0.892·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 1.35·19-s + 0.773·20-s − 0.847·21-s + 1.19·22-s − 0.768·23-s + 0.204·24-s + 1.39·25-s + 0.809·26-s − 0.192·27-s + 0.734·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 3.45T + 5T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 19 | \( 1 + 5.92T + 19T^{2} \) |
| 23 | \( 1 + 3.68T + 23T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 + 6.53T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 0.713T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 - 5.20T + 53T^{2} \) |
| 61 | \( 1 + 0.310T + 61T^{2} \) |
| 67 | \( 1 - 1.06T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 16.5T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 9.13T + 89T^{2} \) |
| 97 | \( 1 + 0.689T + 97T^{2} \) |
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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
−7.903725731157524126303493792352, −7.08657617911812817386106760773, −6.24658044097155075871189061314, −5.53271425371105103290528874261, −5.09633855671678986685319489288, −4.39339228468596145291607692769, −2.60877288594549146136676184640, −2.20774787372446168443460327767, −1.41917824682969714585807009462, 0,
1.41917824682969714585807009462, 2.20774787372446168443460327767, 2.60877288594549146136676184640, 4.39339228468596145291607692769, 5.09633855671678986685319489288, 5.53271425371105103290528874261, 6.24658044097155075871189061314, 7.08657617911812817386106760773, 7.903725731157524126303493792352
