| L(s) = 1 | + 2·5-s − 4·9-s − 4·11-s − 4·13-s + 4·17-s + 2·19-s + 8·23-s + 3·25-s − 4·29-s − 12·37-s + 4·41-s − 8·45-s − 6·49-s − 12·53-s − 8·55-s − 8·61-s − 8·65-s + 16·67-s + 8·71-s − 4·73-s − 24·79-s + 7·81-s − 16·83-s + 8·85-s − 12·89-s + 4·95-s + 4·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 4/3·9-s − 1.20·11-s − 1.10·13-s + 0.970·17-s + 0.458·19-s + 1.66·23-s + 3/5·25-s − 0.742·29-s − 1.97·37-s + 0.624·41-s − 1.19·45-s − 6/7·49-s − 1.64·53-s − 1.07·55-s − 1.02·61-s − 0.992·65-s + 1.95·67-s + 0.949·71-s − 0.468·73-s − 2.70·79-s + 7/9·81-s − 1.75·83-s + 0.867·85-s − 1.27·89-s + 0.410·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 196 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 24 T + 294 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 206 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 180 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.84573085441611604190462506797, −7.55571858546046352382118531745, −7.16886054249533153222984039792, −6.92071211973660388149467515843, −6.49010441233423822092893640020, −5.97301020542692230519175696913, −5.59117010678445213645421387771, −5.46795454153289930333377358772, −5.00086272254816189801690162979, −4.98811605024256293591925753804, −4.37254084842539379744185203764, −3.66778977815518187158393972150, −3.19140975659490072261604385402, −3.01645250443286324263416034929, −2.53847808145534087495016566268, −2.32700556284505036480282290071, −1.47376425097246815246947274121, −1.24008545660311617049862534068, 0, 0,
1.24008545660311617049862534068, 1.47376425097246815246947274121, 2.32700556284505036480282290071, 2.53847808145534087495016566268, 3.01645250443286324263416034929, 3.19140975659490072261604385402, 3.66778977815518187158393972150, 4.37254084842539379744185203764, 4.98811605024256293591925753804, 5.00086272254816189801690162979, 5.46795454153289930333377358772, 5.59117010678445213645421387771, 5.97301020542692230519175696913, 6.49010441233423822092893640020, 6.92071211973660388149467515843, 7.16886054249533153222984039792, 7.55571858546046352382118531745, 7.84573085441611604190462506797
