| L(s) = 1 | − 2·9-s − 2·11-s − 2·19-s + 49-s − 2·61-s + 3·81-s + 4·99-s + 4·101-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 4·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2·9-s − 2·11-s − 2·19-s + 49-s − 2·61-s + 3·81-s + 4·99-s + 4·101-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 4·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4540546875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4540546875\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.509947449414602419106130512195, −9.013980968579495021598026174935, −8.809579887284687448615259578896, −8.364594750470749039077062346468, −8.193609026008999431328878211945, −7.63875873177797577440868630389, −7.48196833966580025376413301999, −6.79472724693543134209112575982, −6.22018702024616175722806379509, −6.02048931963967397356330763431, −5.70871104286145118615489737517, −5.01508359892013033273867357364, −4.98472888269810639769922012330, −4.32457160612193697710962987326, −3.73575783513243868403494954662, −3.10376437912904939628873921552, −2.78733877447800069076163046380, −2.31783756224388070910610549951, −1.90230738564628567184693902590, −0.45808780948722940640938766709,
0.45808780948722940640938766709, 1.90230738564628567184693902590, 2.31783756224388070910610549951, 2.78733877447800069076163046380, 3.10376437912904939628873921552, 3.73575783513243868403494954662, 4.32457160612193697710962987326, 4.98472888269810639769922012330, 5.01508359892013033273867357364, 5.70871104286145118615489737517, 6.02048931963967397356330763431, 6.22018702024616175722806379509, 6.79472724693543134209112575982, 7.48196833966580025376413301999, 7.63875873177797577440868630389, 8.193609026008999431328878211945, 8.364594750470749039077062346468, 8.809579887284687448615259578896, 9.013980968579495021598026174935, 9.509947449414602419106130512195
