| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 2·9-s + 5·16-s − 4·18-s − 8·23-s − 25-s + 12·29-s − 6·32-s + 6·36-s + 4·37-s + 16·46-s + 2·50-s + 4·53-s − 24·58-s + 7·64-s + 8·67-s + 8·71-s − 8·72-s − 8·74-s − 8·79-s − 5·81-s − 24·92-s − 3·100-s − 8·106-s − 24·107-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 2/3·9-s + 5/4·16-s − 0.942·18-s − 1.66·23-s − 1/5·25-s + 2.22·29-s − 1.06·32-s + 36-s + 0.657·37-s + 2.35·46-s + 0.282·50-s + 0.549·53-s − 3.15·58-s + 7/8·64-s + 0.977·67-s + 0.949·71-s − 0.942·72-s − 0.929·74-s − 0.900·79-s − 5/9·81-s − 2.50·92-s − 0.299·100-s − 0.777·106-s − 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8679433700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8679433700\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
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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
−8.891158897028321141827675188911, −8.460507695365043007169966797925, −8.163330158576196973851578599173, −7.67072633424520600901028899031, −7.23345671301248140490078596948, −6.65646240645427821221916998035, −6.31975357162129897913673020661, −5.80553072658033006793774437533, −5.09530992963076420392246102059, −4.37005609544378656310311780982, −3.88210222660779749684404761485, −3.00596767675502013638477250622, −2.36733644359109847388282124042, −1.65668954833119531501770703342, −0.73466411849174535798258787641,
0.73466411849174535798258787641, 1.65668954833119531501770703342, 2.36733644359109847388282124042, 3.00596767675502013638477250622, 3.88210222660779749684404761485, 4.37005609544378656310311780982, 5.09530992963076420392246102059, 5.80553072658033006793774437533, 6.31975357162129897913673020661, 6.65646240645427821221916998035, 7.23345671301248140490078596948, 7.67072633424520600901028899031, 8.163330158576196973851578599173, 8.460507695365043007169966797925, 8.891158897028321141827675188911
