| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s + 8·6-s + 4·7-s + 8·9-s − 16·11-s − 8·12-s + 6·13-s − 8·14-s − 4·16-s + 14·17-s − 16·18-s − 16·21-s + 32·22-s − 4·23-s − 25·25-s − 12·26-s − 36·27-s + 8·28-s + 104·31-s + 8·32-s + 64·33-s − 28·34-s + 16·36-s − 6·37-s − 24·39-s + ⋯ |
| L(s) = 1 | − 2-s − 4/3·3-s + 1/2·4-s + 4/3·6-s + 4/7·7-s + 8/9·9-s − 1.45·11-s − 2/3·12-s + 6/13·13-s − 4/7·14-s − 1/4·16-s + 0.823·17-s − 8/9·18-s − 0.761·21-s + 1.45·22-s − 0.173·23-s − 25-s − 0.461·26-s − 4/3·27-s + 2/7·28-s + 3.35·31-s + 1/4·32-s + 1.93·33-s − 0.823·34-s + 4/9·36-s − 0.162·37-s − 0.615·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2393708110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2393708110\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 52 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 84 T + 3528 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 124 T + 7688 T^{2} - 124 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 94 T + 4418 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9442 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T + 7938 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.97670027424001189851460753414, −20.91840880760687132371699108974, −19.62802241243125925268870039984, −18.93596683106944575370378584993, −18.20751455061888268560137839128, −17.92223562712510451121164587598, −17.06911658410946393747976120432, −16.73674996682997615882681407582, −15.66615912972948546686921550101, −15.36823414016552186873536868229, −13.84524531625142941910940971889, −13.13024352722512341574554202219, −11.76761621055593255088125467995, −11.58377096028888216433546983152, −10.23281628919080119416287605008, −10.14654137527546697435584467874, −8.393364582816363596048703883492, −7.67558497356483516574264525363, −6.19183269878294484670114036539, −5.03540251964772219755791684917,
5.03540251964772219755791684917, 6.19183269878294484670114036539, 7.67558497356483516574264525363, 8.393364582816363596048703883492, 10.14654137527546697435584467874, 10.23281628919080119416287605008, 11.58377096028888216433546983152, 11.76761621055593255088125467995, 13.13024352722512341574554202219, 13.84524531625142941910940971889, 15.36823414016552186873536868229, 15.66615912972948546686921550101, 16.73674996682997615882681407582, 17.06911658410946393747976120432, 17.92223562712510451121164587598, 18.20751455061888268560137839128, 18.93596683106944575370378584993, 19.62802241243125925268870039984, 20.91840880760687132371699108974, 20.97670027424001189851460753414
