| L(s) = 1 | + 2·3-s + 8·7-s + 3·9-s − 6·11-s + 6·13-s + 2·17-s + 7·19-s + 16·21-s + 4·23-s + 10·27-s − 29-s − 10·31-s − 12·33-s + 8·37-s + 12·39-s − 2·41-s + 6·47-s + 34·49-s + 4·51-s − 6·53-s + 14·57-s + 59-s + 7·61-s + 24·63-s − 14·67-s + 8·69-s − 15·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 3.02·7-s + 9-s − 1.80·11-s + 1.66·13-s + 0.485·17-s + 1.60·19-s + 3.49·21-s + 0.834·23-s + 1.92·27-s − 0.185·29-s − 1.79·31-s − 2.08·33-s + 1.31·37-s + 1.92·39-s − 0.312·41-s + 0.875·47-s + 34/7·49-s + 0.560·51-s − 0.824·53-s + 1.85·57-s + 0.130·59-s + 0.896·61-s + 3.02·63-s − 1.71·67-s + 0.963·69-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.083881262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.083881262\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 17 T + 200 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 47 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.185966347256608838486731586542, −8.885271539108395495634580380864, −8.403553996277668332390904360092, −8.262525055877581624746629679982, −8.028011200385701199778456805967, −7.54812121296920486526209381431, −7.20077439042336357277017919332, −7.14660415200485460840162573864, −5.98977008679118641934874384442, −5.69030909251564658771053480452, −5.16408250055960883457152531609, −5.11702550620692209216151035385, −4.35293500856012483902849609656, −4.26836777191941391460572853982, −3.40930429422636265560328362352, −3.02160327713235391472926172150, −2.55875957768212249296486296401, −1.88163309739496665316701301808, −1.35554462950371377971177577490, −1.08844407421819833641327426099,
1.08844407421819833641327426099, 1.35554462950371377971177577490, 1.88163309739496665316701301808, 2.55875957768212249296486296401, 3.02160327713235391472926172150, 3.40930429422636265560328362352, 4.26836777191941391460572853982, 4.35293500856012483902849609656, 5.11702550620692209216151035385, 5.16408250055960883457152531609, 5.69030909251564658771053480452, 5.98977008679118641934874384442, 7.14660415200485460840162573864, 7.20077439042336357277017919332, 7.54812121296920486526209381431, 8.028011200385701199778456805967, 8.262525055877581624746629679982, 8.403553996277668332390904360092, 8.885271539108395495634580380864, 9.185966347256608838486731586542
