| L(s) = 1 | − 3-s + 6·11-s + 4·13-s + 4·19-s + 6·23-s + 5·25-s + 27-s + 12·29-s − 8·31-s − 6·33-s − 2·37-s − 4·39-s + 24·41-s − 8·43-s − 12·47-s + 6·53-s − 4·57-s + 10·61-s − 8·67-s − 6·69-s + 12·71-s + 10·73-s − 5·75-s + 4·79-s − 81-s − 24·83-s − 12·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.80·11-s + 1.10·13-s + 0.917·19-s + 1.25·23-s + 25-s + 0.192·27-s + 2.22·29-s − 1.43·31-s − 1.04·33-s − 0.328·37-s − 0.640·39-s + 3.74·41-s − 1.21·43-s − 1.75·47-s + 0.824·53-s − 0.529·57-s + 1.28·61-s − 0.977·67-s − 0.722·69-s + 1.42·71-s + 1.17·73-s − 0.577·75-s + 0.450·79-s − 1/9·81-s − 2.63·83-s − 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.056066730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.056066730\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 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
−10.87020691644171622528047026403, −10.79094653225598986408776129999, −9.903872468831631807239936662163, −9.675379819704747705888887866276, −9.055002276238311523165184370499, −8.870941719115200052223813108011, −8.367263437544741754394906904257, −7.87656267761473699156152974313, −6.97806787793626094328148046787, −6.97205489038106508013561606991, −6.37666893824194792476706273193, −6.05518776469725952841377631409, −5.28038697610206779737532920156, −5.06811282590913572284262078589, −4.17534176358488821742811419980, −3.92971438854184200707455607170, −3.16765464597929940935755513166, −2.62376870101628358339758927299, −1.23902767568951116073407019843, −1.11439675822313629821117992368,
1.11439675822313629821117992368, 1.23902767568951116073407019843, 2.62376870101628358339758927299, 3.16765464597929940935755513166, 3.92971438854184200707455607170, 4.17534176358488821742811419980, 5.06811282590913572284262078589, 5.28038697610206779737532920156, 6.05518776469725952841377631409, 6.37666893824194792476706273193, 6.97205489038106508013561606991, 6.97806787793626094328148046787, 7.87656267761473699156152974313, 8.367263437544741754394906904257, 8.870941719115200052223813108011, 9.055002276238311523165184370499, 9.675379819704747705888887866276, 9.903872468831631807239936662163, 10.79094653225598986408776129999, 10.87020691644171622528047026403
