| L(s) = 1 | − 2·3-s − 2·5-s + 3·9-s − 4·11-s + 4·13-s + 4·15-s + 2·19-s − 4·23-s + 3·25-s − 4·27-s − 4·31-s + 8·33-s + 4·37-s − 8·39-s + 8·43-s − 6·45-s − 12·47-s − 12·49-s + 8·55-s − 4·57-s − 8·59-s + 8·61-s − 8·65-s + 8·69-s − 24·71-s − 12·73-s − 6·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 9-s − 1.20·11-s + 1.10·13-s + 1.03·15-s + 0.458·19-s − 0.834·23-s + 3/5·25-s − 0.769·27-s − 0.718·31-s + 1.39·33-s + 0.657·37-s − 1.28·39-s + 1.21·43-s − 0.894·45-s − 1.75·47-s − 1.71·49-s + 1.07·55-s − 0.529·57-s − 1.04·59-s + 1.02·61-s − 0.992·65-s + 0.963·69-s − 2.84·71-s − 1.40·73-s − 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5198400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5198400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 224 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 100 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.653101257172932543872864671517, −8.380671298785790573737727161582, −8.021955970496138956400688901841, −7.61198152161155619803094850947, −7.34157288250975547961024115156, −6.91870204079932008294063002951, −6.37429705129694258859340595318, −6.07896679221273678038271922510, −5.61757616777481145003275064796, −5.40488396591193956780985339433, −4.77363508106997912110507763089, −4.47273238523424540724586163258, −4.00300415427074284434623450461, −3.61289088557927660214690512565, −2.91913670267381462709067737859, −2.65141713724831129710305906523, −1.47505248430404790457120522163, −1.36478396081720539859167425152, 0, 0,
1.36478396081720539859167425152, 1.47505248430404790457120522163, 2.65141713724831129710305906523, 2.91913670267381462709067737859, 3.61289088557927660214690512565, 4.00300415427074284434623450461, 4.47273238523424540724586163258, 4.77363508106997912110507763089, 5.40488396591193956780985339433, 5.61757616777481145003275064796, 6.07896679221273678038271922510, 6.37429705129694258859340595318, 6.91870204079932008294063002951, 7.34157288250975547961024115156, 7.61198152161155619803094850947, 8.021955970496138956400688901841, 8.380671298785790573737727161582, 8.653101257172932543872864671517
