| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s + 4·17-s + 2·23-s − 25-s − 4·27-s + 10·29-s − 3·36-s + 22·43-s − 2·48-s + 10·49-s − 8·51-s + 12·53-s + 20·61-s − 64-s − 4·68-s − 4·69-s + 2·75-s − 22·79-s + 5·81-s − 20·87-s − 2·92-s + 100-s − 4·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s + 0.970·17-s + 0.417·23-s − 1/5·25-s − 0.769·27-s + 1.85·29-s − 1/2·36-s + 3.35·43-s − 0.288·48-s + 10/7·49-s − 1.12·51-s + 1.64·53-s + 2.56·61-s − 1/8·64-s − 0.485·68-s − 0.481·69-s + 0.230·75-s − 2.47·79-s + 5/9·81-s − 2.14·87-s − 0.208·92-s + 1/10·100-s − 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.048324105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.048324105\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 107 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.442056504694076169330668583022, −8.120528454565688642950939325201, −7.43305691806305787979894907511, −7.36379449332286867902451332506, −7.04596303977487781270613693294, −6.60284507937044046917999370680, −6.05743375122740498678500479902, −5.87222355214376018319351739419, −5.46792732898408456677176856269, −5.32222379829402757503548059111, −4.68482692401855903216629956485, −4.44352471960726017744804309082, −3.95172934677347510245855299813, −3.79976857185222834521731438940, −3.05037079576461632985435591999, −2.57310560744625012538482499110, −2.24870790747808629568340072805, −1.30012115050580688730725716993, −0.909375857703112440824229733450, −0.58381853848495046382908327236,
0.58381853848495046382908327236, 0.909375857703112440824229733450, 1.30012115050580688730725716993, 2.24870790747808629568340072805, 2.57310560744625012538482499110, 3.05037079576461632985435591999, 3.79976857185222834521731438940, 3.95172934677347510245855299813, 4.44352471960726017744804309082, 4.68482692401855903216629956485, 5.32222379829402757503548059111, 5.46792732898408456677176856269, 5.87222355214376018319351739419, 6.05743375122740498678500479902, 6.60284507937044046917999370680, 7.04596303977487781270613693294, 7.36379449332286867902451332506, 7.43305691806305787979894907511, 8.120528454565688642950939325201, 8.442056504694076169330668583022
