| L(s) = 1 | − 2·2-s + 3·4-s − 4·5-s + 2·7-s − 4·8-s + 8·10-s − 4·14-s + 5·16-s − 4·19-s − 12·20-s + 2·23-s + 2·25-s + 6·28-s + 4·29-s + 4·31-s − 6·32-s − 8·35-s − 4·37-s + 8·38-s + 16·40-s + 12·41-s + 8·43-s − 4·46-s − 4·47-s + 3·49-s − 4·50-s + 4·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.78·5-s + 0.755·7-s − 1.41·8-s + 2.52·10-s − 1.06·14-s + 5/4·16-s − 0.917·19-s − 2.68·20-s + 0.417·23-s + 2/5·25-s + 1.13·28-s + 0.742·29-s + 0.718·31-s − 1.06·32-s − 1.35·35-s − 0.657·37-s + 1.29·38-s + 2.52·40-s + 1.87·41-s + 1.21·43-s − 0.589·46-s − 0.583·47-s + 3/7·49-s − 0.565·50-s + 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9156274393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9156274393\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 14 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.730247826715886651656674922645, −8.670052937494835229885227240065, −8.001045887673468619505422420169, −7.957824570035957710686254924340, −7.60852746896310615468825788603, −7.42182084171457260544066133930, −6.82440993416802819507755564696, −6.51603647682465973212157543356, −5.96967464427714120423892927978, −5.76307538955763884647217413749, −4.78464325415450556113892502830, −4.78277130093845119921641208574, −4.07801953932556335343293690106, −3.92132793546329028207118067492, −3.12997846662315613764326652849, −2.92116632587177739139277256867, −2.03427891459474538975555527101, −1.85535730111492892065131383419, −0.74008259790233754967530449671, −0.59735238308337249023788317737,
0.59735238308337249023788317737, 0.74008259790233754967530449671, 1.85535730111492892065131383419, 2.03427891459474538975555527101, 2.92116632587177739139277256867, 3.12997846662315613764326652849, 3.92132793546329028207118067492, 4.07801953932556335343293690106, 4.78277130093845119921641208574, 4.78464325415450556113892502830, 5.76307538955763884647217413749, 5.96967464427714120423892927978, 6.51603647682465973212157543356, 6.82440993416802819507755564696, 7.42182084171457260544066133930, 7.60852746896310615468825788603, 7.957824570035957710686254924340, 8.001045887673468619505422420169, 8.670052937494835229885227240065, 8.730247826715886651656674922645
