L(s) = 1 | − 3-s + 5-s + 9-s − 2·13-s − 15-s − 6·17-s + 19-s + 4·23-s + 25-s − 27-s − 6·29-s − 2·37-s + 2·39-s − 2·41-s − 4·43-s + 45-s − 4·47-s − 7·49-s + 6·51-s − 6·53-s − 57-s + 12·59-s − 2·61-s − 2·65-s + 4·67-s − 4·69-s − 6·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 1.45·17-s + 0.229·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.328·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s + 0.149·45-s − 0.583·47-s − 49-s + 0.840·51-s − 0.824·53-s − 0.132·57-s + 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.488·67-s − 0.481·69-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.768626210118887679291854915430, −7.78052147180357180110439859452, −6.87606868618011397600478902138, −6.41809015068399926337628047211, −5.33754001761415446732491834460, −4.84409972963554850352386409259, −3.78706787853046917517176527305, −2.59414462363986928995777015747, −1.57225358008099559303589986979, 0,
1.57225358008099559303589986979, 2.59414462363986928995777015747, 3.78706787853046917517176527305, 4.84409972963554850352386409259, 5.33754001761415446732491834460, 6.41809015068399926337628047211, 6.87606868618011397600478902138, 7.78052147180357180110439859452, 8.768626210118887679291854915430