L(s) = 1 | + 3-s + 9-s + 4·11-s + 4·23-s + 2·25-s + 27-s + 4·33-s + 4·37-s − 4·47-s − 2·49-s + 8·59-s − 4·61-s + 4·69-s − 12·71-s + 12·73-s + 2·75-s + 81-s + 4·83-s + 12·97-s + 4·99-s + 8·109-s + 4·111-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.834·23-s + 2/5·25-s + 0.192·27-s + 0.696·33-s + 0.657·37-s − 0.583·47-s − 2/7·49-s + 1.04·59-s − 0.512·61-s + 0.481·69-s − 1.42·71-s + 1.40·73-s + 0.230·75-s + 1/9·81-s + 0.439·83-s + 1.21·97-s + 0.402·99-s + 0.766·109-s + 0.379·111-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.676751655\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.676751655\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.628487900383492598993510374240, −8.204814171542443907722766915591, −7.71173147363065234460053037659, −7.15365050110920796495290191775, −6.86825481356019132853153979925, −6.29249282612145543696541190418, −5.94754963338483809841491324961, −5.15686467512162194059645041475, −4.73635082351087066511838190104, −4.17934691811678675924096182268, −3.61618599282106991251617739631, −3.14542382736284064634413746643, −2.45823839523700293586112297040, −1.69308770289657648500084898332, −0.922965770619143364037465500623,
0.922965770619143364037465500623, 1.69308770289657648500084898332, 2.45823839523700293586112297040, 3.14542382736284064634413746643, 3.61618599282106991251617739631, 4.17934691811678675924096182268, 4.73635082351087066511838190104, 5.15686467512162194059645041475, 5.94754963338483809841491324961, 6.29249282612145543696541190418, 6.86825481356019132853153979925, 7.15365050110920796495290191775, 7.71173147363065234460053037659, 8.204814171542443907722766915591, 8.628487900383492598993510374240