| L(s) = 1 | − 3-s + 3·5-s + 7-s + 9-s − 4·11-s − 3·13-s − 3·15-s − 4·17-s − 21-s − 23-s + 4·25-s − 27-s + 3·29-s + 6·31-s + 4·33-s + 3·35-s − 9·37-s + 3·39-s + 9·41-s + 3·43-s + 3·45-s + 7·47-s + 49-s + 4·51-s − 4·53-s − 12·55-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.832·13-s − 0.774·15-s − 0.970·17-s − 0.218·21-s − 0.208·23-s + 4/5·25-s − 0.192·27-s + 0.557·29-s + 1.07·31-s + 0.696·33-s + 0.507·35-s − 1.47·37-s + 0.480·39-s + 1.40·41-s + 0.457·43-s + 0.447·45-s + 1.02·47-s + 1/7·49-s + 0.560·51-s − 0.549·53-s − 1.61·55-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.39131206325256446041809734207, −6.76424663937870738066321018325, −5.99234362564232113795365538158, −5.45713141654302121882301471663, −4.87107523373367854031150290509, −4.20290822415985263475630282402, −2.70777129824037519230002306274, −2.34259324699707868728412627679, −1.33003807742036272405319904717, 0,
1.33003807742036272405319904717, 2.34259324699707868728412627679, 2.70777129824037519230002306274, 4.20290822415985263475630282402, 4.87107523373367854031150290509, 5.45713141654302121882301471663, 5.99234362564232113795365538158, 6.76424663937870738066321018325, 7.39131206325256446041809734207
