| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 11-s − 15-s + 17-s − 2·21-s + 4·23-s + 25-s − 27-s − 2·29-s − 4·31-s − 33-s + 2·35-s + 2·37-s − 6·43-s + 45-s − 3·49-s − 51-s + 6·53-s + 55-s + 14·59-s + 2·61-s + 2·63-s + 14·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.258·15-s + 0.242·17-s − 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.174·33-s + 0.338·35-s + 0.328·37-s − 0.914·43-s + 0.149·45-s − 3/7·49-s − 0.140·51-s + 0.824·53-s + 0.134·55-s + 1.82·59-s + 0.256·61-s + 0.251·63-s + 1.71·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.30187977699424, −12.91058519066106, −12.50073111448552, −11.91866613149043, −11.39710780584658, −11.12374186841094, −10.74147849539344, −9.924704512560677, −9.809873420612528, −9.182273516265264, −8.507956879364154, −8.277017409783224, −7.548816484069784, −6.987487406919206, −6.720462820090427, −6.027702658347072, −5.436734558243671, −5.175944221016305, −4.665531901041093, −3.885385104819721, −3.570727099398863, −2.613537228078720, −2.140326818001875, −1.378924844273427, −0.9753148494874213, 0,
0.9753148494874213, 1.378924844273427, 2.140326818001875, 2.613537228078720, 3.570727099398863, 3.885385104819721, 4.665531901041093, 5.175944221016305, 5.436734558243671, 6.027702658347072, 6.720462820090427, 6.987487406919206, 7.548816484069784, 8.277017409783224, 8.507956879364154, 9.182273516265264, 9.809873420612528, 9.924704512560677, 10.74147849539344, 11.12374186841094, 11.39710780584658, 11.91866613149043, 12.50073111448552, 12.91058519066106, 13.30187977699424
