L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 4·11-s + 6·12-s + 6·13-s + 5·16-s + 6·17-s − 6·18-s − 8·22-s + 2·23-s − 8·24-s − 12·26-s + 4·27-s + 2·29-s − 2·31-s − 6·32-s + 8·33-s − 12·34-s + 9·36-s + 12·39-s − 6·41-s + 6·43-s + 12·44-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.41·8-s + 9-s + 1.20·11-s + 1.73·12-s + 1.66·13-s + 5/4·16-s + 1.45·17-s − 1.41·18-s − 1.70·22-s + 0.417·23-s − 1.63·24-s − 2.35·26-s + 0.769·27-s + 0.371·29-s − 0.359·31-s − 1.06·32-s + 1.39·33-s − 2.05·34-s + 3/2·36-s + 1.92·39-s − 0.937·41-s + 0.914·43-s + 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.576085698\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.576085698\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 11 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 61 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 93 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 109 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 123 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 160 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 144 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.041555851186913625288322312294, −7.897701697077876066231832078797, −7.44609286087223495405952354758, −7.31193330571970690133007942501, −6.69733594249245796766209872567, −6.61747002678519463358507992263, −6.00354302515484419868841885436, −5.99100082271201054162744445338, −5.19454460082498916006587547631, −5.14136987488770864214114403276, −4.13248206154617541701432238752, −4.02474321001335692229011899624, −3.51982733141073725737441165224, −3.45478792375148122724305579183, −2.65931115796620108641636662602, −2.59448175784176780295541620585, −1.73016443050541720897329165760, −1.57346745670362878041009768221, −0.922714190537144890624526445936, −0.78379502172290067623818240660,
0.78379502172290067623818240660, 0.922714190537144890624526445936, 1.57346745670362878041009768221, 1.73016443050541720897329165760, 2.59448175784176780295541620585, 2.65931115796620108641636662602, 3.45478792375148122724305579183, 3.51982733141073725737441165224, 4.02474321001335692229011899624, 4.13248206154617541701432238752, 5.14136987488770864214114403276, 5.19454460082498916006587547631, 5.99100082271201054162744445338, 6.00354302515484419868841885436, 6.61747002678519463358507992263, 6.69733594249245796766209872567, 7.31193330571970690133007942501, 7.44609286087223495405952354758, 7.897701697077876066231832078797, 8.041555851186913625288322312294