| L(s) = 1 | − 3·3-s − 10·7-s + 6·9-s − 9·13-s − 8·19-s + 30·21-s − 5·25-s − 9·27-s + 27·39-s + 13·43-s + 61·49-s + 24·57-s − 61-s − 60·63-s − 21·67-s − 17·73-s + 15·75-s − 9·79-s + 9·81-s + 90·91-s − 24·97-s − 36·109-s − 54·117-s + 22·121-s + 127-s − 39·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 3.77·7-s + 2·9-s − 2.49·13-s − 1.83·19-s + 6.54·21-s − 25-s − 1.73·27-s + 4.32·39-s + 1.98·43-s + 61/7·49-s + 3.17·57-s − 0.128·61-s − 7.55·63-s − 2.56·67-s − 1.98·73-s + 1.73·75-s − 1.01·79-s + 81-s + 9.43·91-s − 2.43·97-s − 3.44·109-s − 4.99·117-s + 2·121-s + 0.0887·127-s − 3.43·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−12.04201154941349638477142678930, −11.89297722401288184521283510100, −10.66141606790894675708216854284, −10.63449001845275867671211080365, −9.931643418350333875369967605454, −9.842365737202050300264926135144, −9.342262533863547274433536128770, −8.887804814931416945255008589054, −7.57583103918023802549706842705, −7.20942662953002707868174072265, −6.67930668148245683984310915948, −6.44191281389796913815669196940, −5.71299791251531341421742039231, −5.65532936585661186980171838388, −4.28360662784915860997755102290, −4.22478096844127391068925617324, −3.00677681236077986908051424497, −2.49267706559491071174247924841, 0, 0,
2.49267706559491071174247924841, 3.00677681236077986908051424497, 4.22478096844127391068925617324, 4.28360662784915860997755102290, 5.65532936585661186980171838388, 5.71299791251531341421742039231, 6.44191281389796913815669196940, 6.67930668148245683984310915948, 7.20942662953002707868174072265, 7.57583103918023802549706842705, 8.887804814931416945255008589054, 9.342262533863547274433536128770, 9.842365737202050300264926135144, 9.931643418350333875369967605454, 10.63449001845275867671211080365, 10.66141606790894675708216854284, 11.89297722401288184521283510100, 12.04201154941349638477142678930
