| L(s) = 1 | + 4·4-s + 13·7-s − 63·19-s − 25·25-s + 52·28-s + 105·31-s − 73·37-s + 122·43-s + 120·49-s − 168·61-s − 64·64-s + 13·67-s − 189·73-s − 252·76-s − 11·79-s − 100·100-s + 231·103-s + 71·109-s + 121·121-s + 420·124-s + 127-s + 131-s − 819·133-s + 137-s + 139-s − 292·148-s + 149-s + ⋯ |
| L(s) = 1 | + 4-s + 13/7·7-s − 3.31·19-s − 25-s + 13/7·28-s + 3.38·31-s − 1.97·37-s + 2.83·43-s + 2.44·49-s − 2.75·61-s − 64-s + 0.194·67-s − 2.58·73-s − 3.31·76-s − 0.139·79-s − 100-s + 2.24·103-s + 0.651·109-s + 121-s + 3.38·124-s + 0.00787·127-s + 0.00763·131-s − 6.15·133-s + 0.00729·137-s + 0.00719·139-s − 1.97·148-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.837569632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.837569632\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 13 T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )( 1 + 23 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 - 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )( 1 + 47 T + p^{2} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 61 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 47 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 109 T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )( 1 + 143 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
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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
−15.07902367657215131522183328558, −14.55427705800204850411987557067, −13.88619636377177409366953890782, −13.59028325535913707785545531298, −12.48044679524425182942558960747, −12.20905977802725208567214610228, −11.53869866258435274751122380070, −11.09115026222997729412476934948, −10.48528122824025194306920378730, −10.28408700440128297918597752403, −8.802465309030291298696486030989, −8.623334246964728123469544217439, −7.85482144546790696610603288259, −7.29863918295464248640678875541, −6.32906459864325633593013236774, −5.97953310096072569385332632094, −4.49539743087768569403012706023, −4.44364711489392849381763759020, −2.53876036153271695625248551063, −1.80347681501906246688704407860,
1.80347681501906246688704407860, 2.53876036153271695625248551063, 4.44364711489392849381763759020, 4.49539743087768569403012706023, 5.97953310096072569385332632094, 6.32906459864325633593013236774, 7.29863918295464248640678875541, 7.85482144546790696610603288259, 8.623334246964728123469544217439, 8.802465309030291298696486030989, 10.28408700440128297918597752403, 10.48528122824025194306920378730, 11.09115026222997729412476934948, 11.53869866258435274751122380070, 12.20905977802725208567214610228, 12.48044679524425182942558960747, 13.59028325535913707785545531298, 13.88619636377177409366953890782, 14.55427705800204850411987557067, 15.07902367657215131522183328558
