| L(s) = 1 | − 1.12·2-s + 3-s − 0.732·4-s + 2.85·5-s − 1.12·6-s + 7-s + 3.07·8-s + 9-s − 3.21·10-s − 2.08·11-s − 0.732·12-s − 0.785·13-s − 1.12·14-s + 2.85·15-s − 1.99·16-s − 3.99·17-s − 1.12·18-s − 4.47·19-s − 2.09·20-s + 21-s + 2.34·22-s − 3.80·23-s + 3.07·24-s + 3.16·25-s + 0.884·26-s + 27-s − 0.732·28-s + ⋯ |
| L(s) = 1 | − 0.796·2-s + 0.577·3-s − 0.366·4-s + 1.27·5-s − 0.459·6-s + 0.377·7-s + 1.08·8-s + 0.333·9-s − 1.01·10-s − 0.627·11-s − 0.211·12-s − 0.217·13-s − 0.300·14-s + 0.737·15-s − 0.499·16-s − 0.968·17-s − 0.265·18-s − 1.02·19-s − 0.467·20-s + 0.218·21-s + 0.499·22-s − 0.793·23-s + 0.627·24-s + 0.632·25-s + 0.173·26-s + 0.192·27-s − 0.138·28-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 | \( 1 + 1.12T + 2T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 11 | \( 1 + 2.08T + 11T^{2} \) |
| 13 | \( 1 + 0.785T + 13T^{2} \) |
| 17 | \( 1 + 3.99T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 + 3.80T + 23T^{2} \) |
| 29 | \( 1 - 6.83T + 29T^{2} \) |
| 31 | \( 1 + 6.96T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 8.08T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 5.94T + 47T^{2} \) |
| 53 | \( 1 - 7.48T + 53T^{2} \) |
| 59 | \( 1 - 3.28T + 59T^{2} \) |
| 61 | \( 1 + 2.93T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 7.56T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 4.00T + 83T^{2} \) |
| 89 | \( 1 + 8.63T + 89T^{2} \) |
| 97 | \( 1 + 2.52T + 97T^{2} \) |
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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
−7.61994664709872595884142477402, −7.02899652857270665238882347363, −6.09212914183782577672884338113, −5.46330425888628081360519152035, −4.55211585291444282523985376765, −4.09959210383795667910415064580, −2.68744405105665382918969296476, −2.12248702791172381506998298956, −1.37784249431509537366107251106, 0,
1.37784249431509537366107251106, 2.12248702791172381506998298956, 2.68744405105665382918969296476, 4.09959210383795667910415064580, 4.55211585291444282523985376765, 5.46330425888628081360519152035, 6.09212914183782577672884338113, 7.02899652857270665238882347363, 7.61994664709872595884142477402
