| L(s) = 1 | + 4·7-s + 2·9-s + 4·17-s + 12·23-s − 25-s + 8·31-s + 20·41-s − 4·47-s − 2·49-s + 8·63-s + 24·71-s − 20·73-s − 16·79-s − 5·81-s + 12·89-s + 20·97-s − 4·103-s − 12·113-s + 16·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 2/3·9-s + 0.970·17-s + 2.50·23-s − 1/5·25-s + 1.43·31-s + 3.12·41-s − 0.583·47-s − 2/7·49-s + 1.00·63-s + 2.84·71-s − 2.34·73-s − 1.80·79-s − 5/9·81-s + 1.27·89-s + 2.03·97-s − 0.394·103-s − 1.12·113-s + 1.46·119-s + 6/11·121-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 + 0.646·153-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
| \(L(1)\) |
\(\approx\) |
\(3.715136929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.715136929\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−9.776200963334991338265089865452, −9.548542620219817254205937044250, −9.046866256524912021517874461995, −8.513715621080274122741456904179, −8.378850205366672997921174603243, −7.68613461182798119360937015743, −7.42789027444500383537495659354, −7.33199107908077772433713637004, −6.41557191448445407871384308461, −6.31521360589688984426753952956, −5.54024007330661594377694783736, −5.17736275527103570162766340915, −4.71579075765337062072571809757, −4.48420504069430678262865452901, −3.89817626275689786006102587650, −3.17743772980011364424921909151, −2.74380996083893799449863223262, −2.07445675364317249813953668184, −1.14278176642425696447911837790, −1.08571203973696756141475815507,
1.08571203973696756141475815507, 1.14278176642425696447911837790, 2.07445675364317249813953668184, 2.74380996083893799449863223262, 3.17743772980011364424921909151, 3.89817626275689786006102587650, 4.48420504069430678262865452901, 4.71579075765337062072571809757, 5.17736275527103570162766340915, 5.54024007330661594377694783736, 6.31521360589688984426753952956, 6.41557191448445407871384308461, 7.33199107908077772433713637004, 7.42789027444500383537495659354, 7.68613461182798119360937015743, 8.378850205366672997921174603243, 8.513715621080274122741456904179, 9.046866256524912021517874461995, 9.548542620219817254205937044250, 9.776200963334991338265089865452
