| L(s) = 1 | + 2·9-s + 12·17-s − 25-s − 16·31-s + 12·41-s − 24·47-s − 14·49-s + 8·71-s + 20·73-s + 16·79-s − 5·81-s − 28·89-s + 12·97-s − 16·103-s + 4·113-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 2.91·17-s − 1/5·25-s − 2.87·31-s + 1.87·41-s − 3.50·47-s − 2·49-s + 0.949·71-s + 2.34·73-s + 1.80·79-s − 5/9·81-s − 2.96·89-s + 1.21·97-s − 1.57·103-s + 0.376·113-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-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
| \(L(1)\) |
\(\approx\) |
\(2.304515843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.304515843\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−9.816460131092580303730868942707, −9.499169070666755080743966375674, −9.434934158302636330706413414090, −8.428869001446564336675684523140, −8.281400874198584780675458182553, −7.69256064883140028023155388733, −7.62060490790740585039149089420, −7.06159802292879535449280223685, −6.61409216123563902817097371329, −6.10752696829609909612834059768, −5.61183253021110206336438294167, −5.26228028831160479432194375085, −4.93179507836092600893406896408, −4.23081623646308756979369204744, −3.59845659618850537199307707056, −3.41933549884485872791202548018, −2.89836776295036517993281273341, −1.79161102140111418930235135355, −1.63414245439130381187608014171, −0.66125372831476618152163623079,
0.66125372831476618152163623079, 1.63414245439130381187608014171, 1.79161102140111418930235135355, 2.89836776295036517993281273341, 3.41933549884485872791202548018, 3.59845659618850537199307707056, 4.23081623646308756979369204744, 4.93179507836092600893406896408, 5.26228028831160479432194375085, 5.61183253021110206336438294167, 6.10752696829609909612834059768, 6.61409216123563902817097371329, 7.06159802292879535449280223685, 7.62060490790740585039149089420, 7.69256064883140028023155388733, 8.281400874198584780675458182553, 8.428869001446564336675684523140, 9.434934158302636330706413414090, 9.499169070666755080743966375674, 9.816460131092580303730868942707
