| L(s) = 1 | + 4-s + 10·7-s − 2·13-s + 16-s + 10·19-s + 25-s + 10·28-s − 8·31-s − 20·37-s + 16·43-s + 61·49-s − 2·52-s + 16·61-s + 64-s − 8·67-s + 4·73-s + 10·76-s + 4·79-s − 20·91-s − 32·97-s + 100-s − 26·103-s − 32·109-s + 10·112-s − 22·121-s − 8·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 3.77·7-s − 0.554·13-s + 1/4·16-s + 2.29·19-s + 1/5·25-s + 1.88·28-s − 1.43·31-s − 3.28·37-s + 2.43·43-s + 61/7·49-s − 0.277·52-s + 2.04·61-s + 1/8·64-s − 0.977·67-s + 0.468·73-s + 1.14·76-s + 0.450·79-s − 2.09·91-s − 3.24·97-s + 1/10·100-s − 2.56·103-s − 3.06·109-s + 0.944·112-s − 2·121-s − 0.718·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.067697669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.067697669\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 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
−8.388162226041374703892252125543, −7.86566828039851573836004453407, −7.48815220277082266427538573799, −7.12559786726867706746830133866, −6.97971719811016103888145740401, −5.66421127299809088989589629204, −5.46149236623112862096191304232, −5.17636996627577623726694783444, −4.88052625148736655390139569696, −4.07619637962323810583393330436, −3.76718352791364041563854587690, −2.70317199949128469530996564606, −2.20029727063878084208215448489, −1.43718912432457101622262882040, −1.28056241416581044677853894546,
1.28056241416581044677853894546, 1.43718912432457101622262882040, 2.20029727063878084208215448489, 2.70317199949128469530996564606, 3.76718352791364041563854587690, 4.07619637962323810583393330436, 4.88052625148736655390139569696, 5.17636996627577623726694783444, 5.46149236623112862096191304232, 5.66421127299809088989589629204, 6.97971719811016103888145740401, 7.12559786726867706746830133866, 7.48815220277082266427538573799, 7.86566828039851573836004453407, 8.388162226041374703892252125543
