| L(s) = 1 | − 2·3-s + 2·5-s − 2·7-s + 3·9-s + 4·11-s − 4·15-s − 4·17-s + 4·19-s + 4·21-s − 8·23-s + 3·25-s − 4·27-s + 4·29-s − 12·31-s − 8·33-s − 4·35-s − 4·37-s − 4·41-s + 6·45-s − 8·47-s + 3·49-s + 8·51-s + 16·53-s + 8·55-s − 8·57-s + 4·61-s − 6·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s + 1.20·11-s − 1.03·15-s − 0.970·17-s + 0.917·19-s + 0.872·21-s − 1.66·23-s + 3/5·25-s − 0.769·27-s + 0.742·29-s − 2.15·31-s − 1.39·33-s − 0.676·35-s − 0.657·37-s − 0.624·41-s + 0.894·45-s − 1.16·47-s + 3/7·49-s + 1.12·51-s + 2.19·53-s + 1.07·55-s − 1.05·57-s + 0.512·61-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−7.40907161967536259038547280615, −7.37736018875747923426901185698, −6.96836916561908081124979010816, −6.84750611256015935157748459312, −6.17681140360881040882028226030, −6.08434389211386575545212618916, −5.76800999924249683704686821181, −5.54149230808133107004718767107, −5.03865391926768593147842827740, −4.51257713178956453306628118328, −4.32055930098023454324675301693, −3.87414917028970767132927417719, −3.31553872241029480579680288937, −3.17949255512160603970080035580, −2.17598186167833060167337775555, −2.15995041367873390889024500348, −1.30006600228039562552403022399, −1.26881225529082668777199722639, 0, 0,
1.26881225529082668777199722639, 1.30006600228039562552403022399, 2.15995041367873390889024500348, 2.17598186167833060167337775555, 3.17949255512160603970080035580, 3.31553872241029480579680288937, 3.87414917028970767132927417719, 4.32055930098023454324675301693, 4.51257713178956453306628118328, 5.03865391926768593147842827740, 5.54149230808133107004718767107, 5.76800999924249683704686821181, 6.08434389211386575545212618916, 6.17681140360881040882028226030, 6.84750611256015935157748459312, 6.96836916561908081124979010816, 7.37736018875747923426901185698, 7.40907161967536259038547280615
