| L(s) = 1 | + 0.982·2-s − 3-s − 1.03·4-s + 1.89·5-s − 0.982·6-s + 7-s − 2.98·8-s + 9-s + 1.86·10-s − 1.29·11-s + 1.03·12-s − 1.42·13-s + 0.982·14-s − 1.89·15-s − 0.856·16-s + 1.88·17-s + 0.982·18-s − 0.587·19-s − 1.96·20-s − 21-s − 1.27·22-s + 1.50·23-s + 2.98·24-s − 1.40·25-s − 1.40·26-s − 27-s − 1.03·28-s + ⋯ |
| L(s) = 1 | + 0.694·2-s − 0.577·3-s − 0.517·4-s + 0.848·5-s − 0.400·6-s + 0.377·7-s − 1.05·8-s + 0.333·9-s + 0.589·10-s − 0.391·11-s + 0.298·12-s − 0.396·13-s + 0.262·14-s − 0.489·15-s − 0.214·16-s + 0.457·17-s + 0.231·18-s − 0.134·19-s − 0.439·20-s − 0.218·21-s − 0.271·22-s + 0.313·23-s + 0.608·24-s − 0.280·25-s − 0.275·26-s − 0.192·27-s − 0.195·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 0.982T + 2T^{2} \) |
| 5 | \( 1 - 1.89T + 5T^{2} \) |
| 11 | \( 1 + 1.29T + 11T^{2} \) |
| 13 | \( 1 + 1.42T + 13T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 19 | \( 1 + 0.587T + 19T^{2} \) |
| 23 | \( 1 - 1.50T + 23T^{2} \) |
| 29 | \( 1 - 0.717T + 29T^{2} \) |
| 31 | \( 1 + 0.147T + 31T^{2} \) |
| 37 | \( 1 + 5.90T + 37T^{2} \) |
| 41 | \( 1 + 9.28T + 41T^{2} \) |
| 43 | \( 1 - 9.68T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 0.178T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 5.91T + 61T^{2} \) |
| 67 | \( 1 + 2.04T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 3.98T + 73T^{2} \) |
| 79 | \( 1 + 5.96T + 79T^{2} \) |
| 83 | \( 1 - 3.25T + 83T^{2} \) |
| 89 | \( 1 - 1.13T + 89T^{2} \) |
| 97 | \( 1 + 0.396T + 97T^{2} \) |
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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
−7.36675773365048090366579664690, −6.58650600404475864356785261900, −5.67321801004235355041651119690, −5.56228199135076500365539736006, −4.75193892140587439265379871717, −4.14402248264353089926379796967, −3.18329918415118743079267272094, −2.32126735852447721508069455590, −1.27297977113402629028817616778, 0,
1.27297977113402629028817616778, 2.32126735852447721508069455590, 3.18329918415118743079267272094, 4.14402248264353089926379796967, 4.75193892140587439265379871717, 5.56228199135076500365539736006, 5.67321801004235355041651119690, 6.58650600404475864356785261900, 7.36675773365048090366579664690
