L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 3·7-s − 8-s − 2·9-s − 2·10-s − 12-s − 2·13-s − 3·14-s − 2·15-s + 16-s − 2·17-s + 2·18-s + 19-s + 2·20-s − 3·21-s − 23-s + 24-s − 25-s + 2·26-s + 5·27-s + 3·28-s + 5·29-s + 2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s − 2/3·9-s − 0.632·10-s − 0.288·12-s − 0.554·13-s − 0.801·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.471·18-s + 0.229·19-s + 0.447·20-s − 0.654·21-s − 0.208·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.962·27-s + 0.566·28-s + 0.928·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.002052474554601203847041792968, −7.33383284692850104298222628224, −6.45105353442130223755518848132, −5.79961839089031199180500061734, −5.18307333328294993697145102389, −4.44684288371882222822869896351, −3.04609324575723157398341340486, −2.13773804209733337561448475731, −1.39566241001379834421219478911, 0,
1.39566241001379834421219478911, 2.13773804209733337561448475731, 3.04609324575723157398341340486, 4.44684288371882222822869896351, 5.18307333328294993697145102389, 5.79961839089031199180500061734, 6.45105353442130223755518848132, 7.33383284692850104298222628224, 8.002052474554601203847041792968