| L(s) = 1 | − 4·5-s + 7-s − 2·11-s + 6·13-s − 8·17-s − 8·19-s − 2·23-s + 5·25-s + 2·29-s − 4·35-s + 4·37-s + 4·43-s − 12·47-s − 12·53-s + 8·55-s + 8·59-s − 6·61-s − 24·65-s + 8·67-s + 28·71-s − 4·73-s − 2·77-s − 12·79-s + 4·83-s + 32·85-s + 6·91-s + 32·95-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.377·7-s − 0.603·11-s + 1.66·13-s − 1.94·17-s − 1.83·19-s − 0.417·23-s + 25-s + 0.371·29-s − 0.676·35-s + 0.657·37-s + 0.609·43-s − 1.75·47-s − 1.64·53-s + 1.07·55-s + 1.04·59-s − 0.768·61-s − 2.97·65-s + 0.977·67-s + 3.32·71-s − 0.468·73-s − 0.227·77-s − 1.35·79-s + 0.439·83-s + 3.47·85-s + 0.628·91-s + 3.28·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4203032414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4203032414\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.975304893681558232217073111616, −8.513209991261983443747381470336, −8.452445320215308997712389181311, −8.205117874874520281137929420529, −7.83623571249929562897794842347, −7.33160728326287922592009280461, −6.92145705418427138185999443698, −6.48540086436010212859951352585, −6.10929636377027391011311032325, −5.94992491857325362193599234821, −4.89126820607902665139450353310, −4.80409568455309079988321728165, −4.35650923732835223176343893425, −3.90156045537588755392256473343, −3.65201360089004495319850394693, −3.15863311698894899894797157765, −2.19041690873596747963118378836, −2.17742265112025065238118950789, −1.15292812845011734846901312917, −0.24271602992348347599577944285,
0.24271602992348347599577944285, 1.15292812845011734846901312917, 2.17742265112025065238118950789, 2.19041690873596747963118378836, 3.15863311698894899894797157765, 3.65201360089004495319850394693, 3.90156045537588755392256473343, 4.35650923732835223176343893425, 4.80409568455309079988321728165, 4.89126820607902665139450353310, 5.94992491857325362193599234821, 6.10929636377027391011311032325, 6.48540086436010212859951352585, 6.92145705418427138185999443698, 7.33160728326287922592009280461, 7.83623571249929562897794842347, 8.205117874874520281137929420529, 8.452445320215308997712389181311, 8.513209991261983443747381470336, 8.975304893681558232217073111616
