| L(s) = 1 | + 0.482·2-s − 0.300·3-s − 1.76·4-s + 0.848·5-s − 0.144·6-s − 1.74·7-s − 1.81·8-s − 2.90·9-s + 0.409·10-s + 0.172·11-s + 0.530·12-s + 13-s − 0.843·14-s − 0.254·15-s + 2.65·16-s − 7.81·17-s − 1.40·18-s − 7.12·19-s − 1.49·20-s + 0.524·21-s + 0.0832·22-s + 2.39·23-s + 0.546·24-s − 4.28·25-s + 0.482·26-s + 1.77·27-s + 3.08·28-s + ⋯ |
| L(s) = 1 | + 0.341·2-s − 0.173·3-s − 0.883·4-s + 0.379·5-s − 0.0591·6-s − 0.660·7-s − 0.643·8-s − 0.969·9-s + 0.129·10-s + 0.0519·11-s + 0.153·12-s + 0.277·13-s − 0.225·14-s − 0.0657·15-s + 0.663·16-s − 1.89·17-s − 0.331·18-s − 1.63·19-s − 0.335·20-s + 0.114·21-s + 0.0177·22-s + 0.500·23-s + 0.111·24-s − 0.856·25-s + 0.0946·26-s + 0.341·27-s + 0.583·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.482T + 2T^{2} \) |
| 3 | \( 1 + 0.300T + 3T^{2} \) |
| 5 | \( 1 - 0.848T + 5T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 - 0.172T + 11T^{2} \) |
| 17 | \( 1 + 7.81T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 37 | \( 1 + 0.630T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 1.81T + 43T^{2} \) |
| 47 | \( 1 - 1.80T + 47T^{2} \) |
| 53 | \( 1 + 1.19T + 53T^{2} \) |
| 59 | \( 1 - 9.16T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 5.84T + 67T^{2} \) |
| 71 | \( 1 - 2.22T + 71T^{2} \) |
| 73 | \( 1 + 6.64T + 73T^{2} \) |
| 79 | \( 1 + 0.527T + 79T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 0.368T + 97T^{2} \) |
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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
−10.81802565437829790649441826833, −9.859098402759306244550152832849, −8.820812524337246866330274233523, −8.464595160018949820176891713717, −6.64317893000343521001371337420, −6.03382729112380228889567579458, −4.88745098211830776598720783277, −3.87371325859849779412001030455, −2.50978804707082711516881848957, 0,
2.50978804707082711516881848957, 3.87371325859849779412001030455, 4.88745098211830776598720783277, 6.03382729112380228889567579458, 6.64317893000343521001371337420, 8.464595160018949820176891713717, 8.820812524337246866330274233523, 9.859098402759306244550152832849, 10.81802565437829790649441826833
