L(s) = 1 | − 3-s + 3·5-s + 3·9-s − 3·11-s − 4·13-s − 3·15-s + 3·17-s + 19-s + 3·23-s + 5·25-s − 8·27-s − 12·29-s + 7·31-s + 3·33-s + 37-s + 4·39-s − 12·41-s + 8·43-s + 9·45-s + 9·47-s − 3·51-s − 3·53-s − 9·55-s − 57-s − 9·59-s − 61-s − 12·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s + 9-s − 0.904·11-s − 1.10·13-s − 0.774·15-s + 0.727·17-s + 0.229·19-s + 0.625·23-s + 25-s − 1.53·27-s − 2.22·29-s + 1.25·31-s + 0.522·33-s + 0.164·37-s + 0.640·39-s − 1.87·41-s + 1.21·43-s + 1.34·45-s + 1.31·47-s − 0.420·51-s − 0.412·53-s − 1.21·55-s − 0.132·57-s − 1.17·59-s − 0.128·61-s − 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.842263128\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.842263128\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.44935051711723392699614595410, −9.916376550706979719229660705969, −9.909077189914488117649018297796, −9.293682340537926849137024674493, −9.094111790654333707311549372934, −8.408598150886657313185392215239, −7.62926231066195278646976716819, −7.38405865827004440009669189840, −7.32843964470902179094384314182, −6.36785745266162731248624384606, −6.12400464123138238829553209446, −5.56479281380301108940431307860, −5.27620547862612895005068209007, −4.78576284752321927715712470940, −4.34383767953976322852569964045, −3.44333869827717705299929632929, −2.96698847671378968283122958177, −2.01699466734284284737264425041, −1.88021410095585668506948943359, −0.70481772857522623718993770402,
0.70481772857522623718993770402, 1.88021410095585668506948943359, 2.01699466734284284737264425041, 2.96698847671378968283122958177, 3.44333869827717705299929632929, 4.34383767953976322852569964045, 4.78576284752321927715712470940, 5.27620547862612895005068209007, 5.56479281380301108940431307860, 6.12400464123138238829553209446, 6.36785745266162731248624384606, 7.32843964470902179094384314182, 7.38405865827004440009669189840, 7.62926231066195278646976716819, 8.408598150886657313185392215239, 9.094111790654333707311549372934, 9.293682340537926849137024674493, 9.909077189914488117649018297796, 9.916376550706979719229660705969, 10.44935051711723392699614595410