| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s + 8·23-s − 25-s − 4·27-s − 3·36-s + 4·43-s − 2·48-s + 5·49-s − 26·53-s − 4·61-s − 64-s − 16·69-s + 2·75-s − 20·79-s + 5·81-s − 8·92-s + 100-s − 8·101-s + 18·103-s − 12·107-s + 4·108-s + 32·113-s + 21·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s + 1.66·23-s − 1/5·25-s − 0.769·27-s − 1/2·36-s + 0.609·43-s − 0.288·48-s + 5/7·49-s − 3.57·53-s − 0.512·61-s − 1/8·64-s − 1.92·69-s + 0.230·75-s − 2.25·79-s + 5/9·81-s − 0.834·92-s + 1/10·100-s − 0.796·101-s + 1.77·103-s − 1.16·107-s + 0.384·108-s + 3.01·113-s + 1.90·121-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
| \(L(1)\) |
\(\approx\) |
\(0.5233194886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5233194886\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 177 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.624455572400189252113725503015, −7.890366130486890043551756101203, −7.67582679953680771177977052102, −7.28116974565876885695440782314, −6.99872038361984360056429601323, −6.51242010608576055860759163538, −6.22694060213884888219550318361, −5.85568131055190993830714123800, −5.54170786465685025551120874055, −5.09009880763016517780845986568, −4.64830485734619457871118172859, −4.61578999069602632734149920491, −4.10920311006230030776778162191, −3.38394702273183657784340783046, −3.31062367643283639200029230308, −2.64017374191209443633296919649, −2.08655138042056272995658128936, −1.26677341228616871085555583614, −1.18608190003359032364789492095, −0.24066229365829699213152901968,
0.24066229365829699213152901968, 1.18608190003359032364789492095, 1.26677341228616871085555583614, 2.08655138042056272995658128936, 2.64017374191209443633296919649, 3.31062367643283639200029230308, 3.38394702273183657784340783046, 4.10920311006230030776778162191, 4.61578999069602632734149920491, 4.64830485734619457871118172859, 5.09009880763016517780845986568, 5.54170786465685025551120874055, 5.85568131055190993830714123800, 6.22694060213884888219550318361, 6.51242010608576055860759163538, 6.99872038361984360056429601323, 7.28116974565876885695440782314, 7.67582679953680771177977052102, 7.890366130486890043551756101203, 8.624455572400189252113725503015
