| L(s) = 1 | + 2·3-s + 2·7-s + 2·9-s − 8·11-s + 6·13-s + 6·17-s + 4·21-s + 6·23-s + 6·27-s + 4·29-s − 16·33-s + 6·37-s + 12·39-s + 12·41-s − 6·43-s + 18·47-s + 2·49-s + 12·51-s − 10·53-s + 4·63-s − 18·67-s + 12·69-s − 10·73-s − 16·77-s + 11·81-s − 6·83-s + 8·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 2/3·9-s − 2.41·11-s + 1.66·13-s + 1.45·17-s + 0.872·21-s + 1.25·23-s + 1.15·27-s + 0.742·29-s − 2.78·33-s + 0.986·37-s + 1.92·39-s + 1.87·41-s − 0.914·43-s + 2.62·47-s + 2/7·49-s + 1.68·51-s − 1.37·53-s + 0.503·63-s − 2.19·67-s + 1.44·69-s − 1.17·73-s − 1.82·77-s + 11/9·81-s − 0.658·83-s + 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.558060843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.558060843\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.335471871715548963134810547260, −9.199986036096361040644173445779, −8.710792514662502558128758756072, −8.379833289083604216675323831739, −7.957129164992029050136585001242, −7.84634544188999616789983521419, −7.37277993114567478708472277995, −7.11121889197624676284593511399, −6.18081979363828298268674060771, −6.00074081911287784179597347611, −5.42686353838499235316846106494, −5.14303012029710053545229930093, −4.42221592988129295533729913592, −4.32105652789930491658332492539, −3.37673221799184133562580383159, −3.04649248551751460664454761990, −2.76498067237927063748406916396, −2.22009061319121592116103461872, −1.32389948059462705641449834747, −0.868681328818889429323508678969,
0.868681328818889429323508678969, 1.32389948059462705641449834747, 2.22009061319121592116103461872, 2.76498067237927063748406916396, 3.04649248551751460664454761990, 3.37673221799184133562580383159, 4.32105652789930491658332492539, 4.42221592988129295533729913592, 5.14303012029710053545229930093, 5.42686353838499235316846106494, 6.00074081911287784179597347611, 6.18081979363828298268674060771, 7.11121889197624676284593511399, 7.37277993114567478708472277995, 7.84634544188999616789983521419, 7.957129164992029050136585001242, 8.379833289083604216675323831739, 8.710792514662502558128758756072, 9.199986036096361040644173445779, 9.335471871715548963134810547260
