| L(s) = 1 | − 2-s + 4-s − 8-s + 13-s + 16-s − 5·19-s − 26-s − 3·29-s + 4·31-s − 32-s + 7·37-s + 5·38-s + 3·41-s − 2·43-s − 9·47-s + 52-s + 9·53-s + 3·58-s + 6·59-s − 8·61-s − 4·62-s + 64-s − 5·67-s + 3·71-s − 4·73-s − 7·74-s − 5·76-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.277·13-s + 1/4·16-s − 1.14·19-s − 0.196·26-s − 0.557·29-s + 0.718·31-s − 0.176·32-s + 1.15·37-s + 0.811·38-s + 0.468·41-s − 0.304·43-s − 1.31·47-s + 0.138·52-s + 1.23·53-s + 0.393·58-s + 0.781·59-s − 1.02·61-s − 0.508·62-s + 1/8·64-s − 0.610·67-s + 0.356·71-s − 0.468·73-s − 0.813·74-s − 0.573·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.641663191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.641663191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.70465443164665, −12.20291953317871, −11.68088463116978, −11.32416022470260, −10.83646471319504, −10.41208653366560, −9.967388302324030, −9.514952001405982, −8.981158375110071, −8.582207924648481, −8.173151483246972, −7.609484041613832, −7.283528611141105, −6.503898743727588, −6.286948269698491, −5.818344961126768, −5.083318311087110, −4.585018557851692, −4.026151695719317, −3.443882326922550, −2.861992059838521, −2.194406629478461, −1.823560019466532, −0.9802278959181047, −0.4429779539889366,
0.4429779539889366, 0.9802278959181047, 1.823560019466532, 2.194406629478461, 2.861992059838521, 3.443882326922550, 4.026151695719317, 4.585018557851692, 5.083318311087110, 5.818344961126768, 6.286948269698491, 6.503898743727588, 7.283528611141105, 7.609484041613832, 8.173151483246972, 8.582207924648481, 8.981158375110071, 9.514952001405982, 9.967388302324030, 10.41208653366560, 10.83646471319504, 11.32416022470260, 11.68088463116978, 12.20291953317871, 12.70465443164665
