| L(s) = 1 | + 3.16·3-s − 3.16·7-s + 7.00·9-s − 6·13-s + 2·17-s − 6.32·19-s − 10.0·21-s − 3.16·23-s + 12.6·27-s − 4·29-s + 6.32·31-s − 2·37-s − 18.9·39-s − 3.16·43-s + 9.48·47-s + 3.00·49-s + 6.32·51-s − 6·53-s − 20.0·57-s + 6.32·59-s + 2·61-s − 22.1·63-s − 9.48·67-s − 10.0·69-s + 6.32·71-s − 14·73-s − 12.6·79-s + ⋯ |
| L(s) = 1 | + 1.82·3-s − 1.19·7-s + 2.33·9-s − 1.66·13-s + 0.485·17-s − 1.45·19-s − 2.18·21-s − 0.659·23-s + 2.43·27-s − 0.742·29-s + 1.13·31-s − 0.328·37-s − 3.03·39-s − 0.482·43-s + 1.38·47-s + 0.428·49-s + 0.885·51-s − 0.824·53-s − 2.64·57-s + 0.823·59-s + 0.256·61-s − 2.78·63-s − 1.15·67-s − 1.20·69-s + 0.750·71-s − 1.63·73-s − 1.42·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.16T + 3T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6.32T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 9.48T + 67T^{2} \) |
| 71 | \( 1 - 6.32T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.79323290905092300891501689215, −7.07844158460317780517031262481, −6.58656879251271665953127756120, −5.54132867928552008382771591894, −4.38300242345956917613477779296, −3.95355005747643954291079689042, −2.91949215796622178761581626904, −2.61767018852454892943591592025, −1.69844984601260429028066229861, 0,
1.69844984601260429028066229861, 2.61767018852454892943591592025, 2.91949215796622178761581626904, 3.95355005747643954291079689042, 4.38300242345956917613477779296, 5.54132867928552008382771591894, 6.58656879251271665953127756120, 7.07844158460317780517031262481, 7.79323290905092300891501689215
