| L(s) = 1 | − 2·3-s + 3·9-s + 12·17-s − 8·19-s + 2·25-s − 4·27-s + 12·41-s − 8·43-s − 2·49-s − 24·51-s + 16·57-s + 24·59-s + 8·67-s − 4·73-s − 4·75-s + 5·81-s − 12·89-s − 4·97-s + 24·107-s − 12·113-s − 22·121-s − 24·123-s + 127-s + 16·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 2.91·17-s − 1.83·19-s + 2/5·25-s − 0.769·27-s + 1.87·41-s − 1.21·43-s − 2/7·49-s − 3.36·51-s + 2.11·57-s + 3.12·59-s + 0.977·67-s − 0.468·73-s − 0.461·75-s + 5/9·81-s − 1.27·89-s − 0.406·97-s + 2.32·107-s − 1.12·113-s − 2·121-s − 2.16·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.288920275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288920275\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−10.40832323178950496516690073363, −10.24142541407562571412619631750, −9.757942827002881799281795958243, −9.567824670349199995278608367568, −8.600997925327965764229099302844, −8.546702118283194580045683006223, −7.79344604086434859189305136321, −7.60853996407630508849648715281, −6.89673040000006302638164559861, −6.63629104888727364493423594851, −6.01226049225962676935830219852, −5.67974700076737418286177162481, −5.27210138874681810345572871230, −4.82532112152912354925956242158, −4.06798219088688365647526235101, −3.79580701790612401048742900272, −3.02866888416199003373007631606, −2.28313200304748551834209679982, −1.39673240493110064978376001926, −0.68078562470568089929410367530,
0.68078562470568089929410367530, 1.39673240493110064978376001926, 2.28313200304748551834209679982, 3.02866888416199003373007631606, 3.79580701790612401048742900272, 4.06798219088688365647526235101, 4.82532112152912354925956242158, 5.27210138874681810345572871230, 5.67974700076737418286177162481, 6.01226049225962676935830219852, 6.63629104888727364493423594851, 6.89673040000006302638164559861, 7.60853996407630508849648715281, 7.79344604086434859189305136321, 8.546702118283194580045683006223, 8.600997925327965764229099302844, 9.567824670349199995278608367568, 9.757942827002881799281795958243, 10.24142541407562571412619631750, 10.40832323178950496516690073363
