| L(s) = 1 | − 4-s − 2·7-s + 8·13-s + 16-s − 2·25-s + 2·28-s + 8·31-s + 12·37-s + 20·43-s + 3·49-s − 8·52-s + 4·61-s − 64-s + 8·67-s + 13·73-s + 16·79-s − 16·91-s + 4·97-s + 2·100-s − 8·103-s − 12·109-s − 2·112-s − 18·121-s − 8·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.755·7-s + 2.21·13-s + 1/4·16-s − 2/5·25-s + 0.377·28-s + 1.43·31-s + 1.97·37-s + 3.04·43-s + 3/7·49-s − 1.10·52-s + 0.512·61-s − 1/8·64-s + 0.977·67-s + 1.52·73-s + 1.80·79-s − 1.67·91-s + 0.406·97-s + 1/5·100-s − 0.788·103-s − 1.14·109-s − 0.188·112-s − 1.63·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1158948 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1158948 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.103150558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103150558\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 14 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.103098619233242840593532989184, −7.78066763291057376932457012527, −7.17798727368974979686318822900, −6.49300223449028653285865996600, −6.27942522726260046183272711593, −5.98458452061689927476785484740, −5.47980394922742865581936977217, −4.87854429529578469455797236697, −4.24323550317964433133944467004, −3.79501884726705358055623041787, −3.66585800400129297866312155773, −2.71109017621754591930405225987, −2.41367252406877819691446069392, −1.18291351540065578327690139424, −0.794524985258601206646934844449,
0.794524985258601206646934844449, 1.18291351540065578327690139424, 2.41367252406877819691446069392, 2.71109017621754591930405225987, 3.66585800400129297866312155773, 3.79501884726705358055623041787, 4.24323550317964433133944467004, 4.87854429529578469455797236697, 5.47980394922742865581936977217, 5.98458452061689927476785484740, 6.27942522726260046183272711593, 6.49300223449028653285865996600, 7.17798727368974979686318822900, 7.78066763291057376932457012527, 8.103098619233242840593532989184
