| L(s) = 1 | + 3-s + 10·5-s − 14·7-s + 9·9-s + 13·11-s − 19·13-s + 10·15-s + 29·17-s − 14·21-s + 75·25-s + 26·27-s − 23·29-s + 13·33-s − 140·35-s − 19·39-s + 90·45-s − 31·47-s + 147·49-s + 29·51-s + 130·55-s − 126·63-s − 190·65-s + 4·71-s + 68·73-s + 75·75-s − 182·77-s + 157·79-s + ⋯ |
| L(s) = 1 | + 1/3·3-s + 2·5-s − 2·7-s + 9-s + 1.18·11-s − 1.46·13-s + 2/3·15-s + 1.70·17-s − 2/3·21-s + 3·25-s + 0.962·27-s − 0.793·29-s + 0.393·33-s − 4·35-s − 0.487·39-s + 2·45-s − 0.659·47-s + 3·49-s + 0.568·51-s + 2.36·55-s − 2·63-s − 2.92·65-s + 4/71·71-s + 0.931·73-s + 75-s − 2.36·77-s + 1.98·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.595861705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.595861705\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 8 T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T + 48 T^{2} - 13 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 19 T + 192 T^{2} + 19 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 29 T + 552 T^{2} - 29 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23 T - 312 T^{2} + 23 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 31 T - 1248 T^{2} + 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 157 T + 18408 T^{2} - 157 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 86 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 149 T + 12792 T^{2} - 149 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−13.08905915577880214898817369674, −12.63338452469970669398434337529, −12.49045629737800870693463689101, −11.97945549144279487133240824324, −10.93700501956721322053198447846, −10.12009130905570486163942832685, −9.872066313397811908722265563294, −9.851885496310535006602333189589, −9.194259120636629720128679637960, −8.886526363643839169183216869095, −7.70831380635089071256870011567, −7.03166622037096669318874469686, −6.69319261060412453216203409954, −6.14235631764396958234671568657, −5.55210843029910927898706161129, −4.85439546259380135318758172838, −3.76192328071228749097072795932, −3.06907972894607140089878949561, −2.28496982487583331243431496649, −1.19084076574701911907234701732,
1.19084076574701911907234701732, 2.28496982487583331243431496649, 3.06907972894607140089878949561, 3.76192328071228749097072795932, 4.85439546259380135318758172838, 5.55210843029910927898706161129, 6.14235631764396958234671568657, 6.69319261060412453216203409954, 7.03166622037096669318874469686, 7.70831380635089071256870011567, 8.886526363643839169183216869095, 9.194259120636629720128679637960, 9.851885496310535006602333189589, 9.872066313397811908722265563294, 10.12009130905570486163942832685, 10.93700501956721322053198447846, 11.97945549144279487133240824324, 12.49045629737800870693463689101, 12.63338452469970669398434337529, 13.08905915577880214898817369674
