| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 9-s + 2·10-s − 2·12-s − 15-s − 4·16-s + 2·18-s + 8·19-s + 2·20-s − 4·23-s + 25-s − 27-s + 6·29-s − 2·30-s − 8·32-s + 2·36-s + 16·38-s + 45-s − 8·46-s + 20·47-s + 4·48-s − 2·49-s + 2·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s + 0.632·10-s − 0.577·12-s − 0.258·15-s − 16-s + 0.471·18-s + 1.83·19-s + 0.447·20-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.365·30-s − 1.41·32-s + 1/3·36-s + 2.59·38-s + 0.149·45-s − 1.17·46-s + 2.91·47-s + 0.577·48-s − 2/7·49-s + 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.232269705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.232269705\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.073530912643278889469956898388, −8.657861741242126463766634676427, −7.88654228264781158140020591353, −7.47252738528849922863843022880, −6.81868368955996367238914359276, −6.58666760429134442590423229824, −5.82490822183852211395449671228, −5.45856204156854412166855237097, −5.35311022889694298747997708124, −4.33995178638783517592705153269, −4.26296913019575463751863880198, −3.37165696296968462837210908940, −2.83061317783178071719510045796, −2.11884394608975336905441282615, −0.977335489126259654610215845871,
0.977335489126259654610215845871, 2.11884394608975336905441282615, 2.83061317783178071719510045796, 3.37165696296968462837210908940, 4.26296913019575463751863880198, 4.33995178638783517592705153269, 5.35311022889694298747997708124, 5.45856204156854412166855237097, 5.82490822183852211395449671228, 6.58666760429134442590423229824, 6.81868368955996367238914359276, 7.47252738528849922863843022880, 7.88654228264781158140020591353, 8.657861741242126463766634676427, 9.073530912643278889469956898388
