| L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 7·13-s + 3·17-s − 19-s − 21-s + 4·23-s − 27-s − 8·29-s − 6·31-s + 33-s + 7·37-s − 7·39-s − 9·41-s − 6·43-s + 2·47-s + 49-s − 3·51-s + 3·53-s + 57-s + 14·59-s − 3·61-s + 63-s − 5·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.94·13-s + 0.727·17-s − 0.229·19-s − 0.218·21-s + 0.834·23-s − 0.192·27-s − 1.48·29-s − 1.07·31-s + 0.174·33-s + 1.15·37-s − 1.12·39-s − 1.40·41-s − 0.914·43-s + 0.291·47-s + 1/7·49-s − 0.420·51-s + 0.412·53-s + 0.132·57-s + 1.82·59-s − 0.384·61-s + 0.125·63-s − 0.610·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.376999262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.376999262\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.61959443003241, −11.89870318994087, −11.46772690887382, −11.18651245725674, −10.83591542370758, −10.33181500131491, −9.894008754591832, −9.267508936145016, −8.877893643053314, −8.340552395340641, −8.020669435625334, −7.371895921238356, −6.977085798648079, −6.442906247028624, −5.900720694280754, −5.501049618136210, −5.193413506753086, −4.509956011737354, −3.836579046893385, −3.607496569150947, −3.008544782883084, −2.149967783750685, −1.600319135639052, −1.116778874496764, −0.4512023749204111,
0.4512023749204111, 1.116778874496764, 1.600319135639052, 2.149967783750685, 3.008544782883084, 3.607496569150947, 3.836579046893385, 4.509956011737354, 5.193413506753086, 5.501049618136210, 5.900720694280754, 6.442906247028624, 6.977085798648079, 7.371895921238356, 8.020669435625334, 8.340552395340641, 8.877893643053314, 9.267508936145016, 9.894008754591832, 10.33181500131491, 10.83591542370758, 11.18651245725674, 11.46772690887382, 11.89870318994087, 12.61959443003241
