| L(s) = 1 | − 4-s − 4·5-s + 2·7-s + 4·11-s + 12·13-s + 16-s − 2·17-s + 4·20-s − 6·23-s + 8·25-s − 2·28-s + 4·29-s − 6·31-s − 8·35-s − 12·37-s − 2·41-s − 4·44-s + 20·47-s + 2·49-s − 12·52-s − 16·55-s + 4·61-s − 64-s − 48·65-s + 8·67-s + 2·68-s + 2·71-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.78·5-s + 0.755·7-s + 1.20·11-s + 3.32·13-s + 1/4·16-s − 0.485·17-s + 0.894·20-s − 1.25·23-s + 8/5·25-s − 0.377·28-s + 0.742·29-s − 1.07·31-s − 1.35·35-s − 1.97·37-s − 0.312·41-s − 0.603·44-s + 2.91·47-s + 2/7·49-s − 1.66·52-s − 2.15·55-s + 0.512·61-s − 1/8·64-s − 5.95·65-s + 0.977·67-s + 0.242·68-s + 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.227150483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.227150483\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−11.82699567271427670530295196675, −11.53383237076331353896346265809, −10.99204063501587875169903683478, −10.78582451079648955639632056849, −10.34135022542770929091647528781, −9.349288390015091112167286375674, −8.817969526946263001489372439612, −8.656085921386729275390739070308, −8.205742247527223305275223905593, −7.915371461903695213301966083299, −6.93495799039108633180876062431, −6.85895005253238785245726957744, −5.84082181817088696933743350089, −5.67921479878309132841481648321, −4.50790479011145719822752276294, −4.15050813176063097862601971021, −3.65017507812976024460755029739, −3.49331243258603852155308560096, −1.80396497023660261662572338557, −0.919457433327447495734086435265,
0.919457433327447495734086435265, 1.80396497023660261662572338557, 3.49331243258603852155308560096, 3.65017507812976024460755029739, 4.15050813176063097862601971021, 4.50790479011145719822752276294, 5.67921479878309132841481648321, 5.84082181817088696933743350089, 6.85895005253238785245726957744, 6.93495799039108633180876062431, 7.915371461903695213301966083299, 8.205742247527223305275223905593, 8.656085921386729275390739070308, 8.817969526946263001489372439612, 9.349288390015091112167286375674, 10.34135022542770929091647528781, 10.78582451079648955639632056849, 10.99204063501587875169903683478, 11.53383237076331353896346265809, 11.82699567271427670530295196675
