| L(s) = 1 | + 4·3-s − 4·5-s + 6·9-s − 8·13-s − 16·15-s + 11·25-s − 4·27-s − 8·37-s − 32·39-s − 4·41-s + 12·43-s − 24·45-s + 10·49-s − 8·53-s + 32·65-s − 28·67-s + 16·71-s + 44·75-s + 32·79-s − 37·81-s + 4·83-s + 12·89-s − 20·107-s − 32·111-s − 48·117-s + 6·121-s − 16·123-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s + 2·9-s − 2.21·13-s − 4.13·15-s + 11/5·25-s − 0.769·27-s − 1.31·37-s − 5.12·39-s − 0.624·41-s + 1.82·43-s − 3.57·45-s + 10/7·49-s − 1.09·53-s + 3.96·65-s − 3.42·67-s + 1.89·71-s + 5.08·75-s + 3.60·79-s − 4.11·81-s + 0.439·83-s + 1.27·89-s − 1.93·107-s − 3.03·111-s − 4.43·117-s + 6/11·121-s − 1.44·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.722474975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.722474975\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.512218371182951573576753475225, −9.197084419764676463655204011125, −9.180494455103817805734306839866, −8.591262282401732317022432716569, −8.200925432549830308994829089880, −7.75712121214731564645963326270, −7.71642686350118672014726087530, −7.29988577985083121264977916064, −6.97003929060344258222343922688, −6.32625140809115171331283741203, −5.55337239052502009125804132707, −4.95464885666485690105618443325, −4.67939603112629430874142962033, −3.94880278033456316785246763742, −3.75965242751613963035974419456, −3.21124637443676574041708568434, −2.76404632955529745201695174867, −2.42079975713106507910109817716, −1.81354972869085902912592111415, −0.44969418673227156163199383703,
0.44969418673227156163199383703, 1.81354972869085902912592111415, 2.42079975713106507910109817716, 2.76404632955529745201695174867, 3.21124637443676574041708568434, 3.75965242751613963035974419456, 3.94880278033456316785246763742, 4.67939603112629430874142962033, 4.95464885666485690105618443325, 5.55337239052502009125804132707, 6.32625140809115171331283741203, 6.97003929060344258222343922688, 7.29988577985083121264977916064, 7.71642686350118672014726087530, 7.75712121214731564645963326270, 8.200925432549830308994829089880, 8.591262282401732317022432716569, 9.180494455103817805734306839866, 9.197084419764676463655204011125, 9.512218371182951573576753475225
