| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 11-s + 13-s − 15-s + 17-s − 19-s − 2·21-s + 25-s − 27-s − 8·29-s − 9·31-s − 33-s + 2·35-s + 37-s − 39-s + 43-s + 45-s + 2·47-s − 3·49-s − 51-s + 6·53-s + 55-s + 57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.229·19-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.48·29-s − 1.61·31-s − 0.174·33-s + 0.338·35-s + 0.164·37-s − 0.160·39-s + 0.152·43-s + 0.149·45-s + 0.291·47-s − 3/7·49-s − 0.140·51-s + 0.824·53-s + 0.134·55-s + 0.132·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.57052380427191, −14.90401468678786, −14.52706496713085, −14.08744119715671, −13.21891396374703, −13.00625222212288, −12.33790878898536, −11.63787678395078, −11.31641172295410, −10.70632897795947, −10.32181358054326, −9.419308225931892, −9.181004489042801, −8.408081351650167, −7.744033820107841, −7.229603335159161, −6.577951058622641, −5.895941840055892, −5.403135172551159, −4.932859915186031, −4.018042014166831, −3.624943555818707, −2.482296921174677, −1.779378858644760, −1.159053858076272, 0,
1.159053858076272, 1.779378858644760, 2.482296921174677, 3.624943555818707, 4.018042014166831, 4.932859915186031, 5.403135172551159, 5.895941840055892, 6.577951058622641, 7.229603335159161, 7.744033820107841, 8.408081351650167, 9.181004489042801, 9.419308225931892, 10.32181358054326, 10.70632897795947, 11.31641172295410, 11.63787678395078, 12.33790878898536, 13.00625222212288, 13.21891396374703, 14.08744119715671, 14.52706496713085, 14.90401468678786, 15.57052380427191
