| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s + 16-s + 12·23-s − 25-s + 4·27-s − 3·36-s + 8·43-s + 2·48-s + 10·49-s − 12·53-s + 28·61-s − 64-s + 24·69-s − 2·75-s − 32·79-s + 5·81-s − 12·92-s + 100-s + 24·101-s + 32·103-s − 24·107-s − 4·108-s + 24·113-s + 22·121-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1/4·16-s + 2.50·23-s − 1/5·25-s + 0.769·27-s − 1/2·36-s + 1.21·43-s + 0.288·48-s + 10/7·49-s − 1.64·53-s + 3.58·61-s − 1/8·64-s + 2.88·69-s − 0.230·75-s − 3.60·79-s + 5/9·81-s − 1.25·92-s + 1/10·100-s + 2.38·101-s + 3.15·103-s − 2.32·107-s − 0.384·108-s + 2.25·113-s + 2·121-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
| \(L(1)\) |
\(\approx\) |
\(5.365519084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.365519084\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−8.440136194256352642617783361755, −8.322044087526028216359009444055, −7.55158769064946380971256046966, −7.39285385258005308466497507102, −7.01647883510405504944266751933, −6.98598071186865353445104349553, −6.09055078613316102753393896897, −5.99996789858576194144210721437, −5.44178023611126331662578335000, −4.99035317855142227504260741965, −4.71530969459032516177177655947, −4.27401694018432444927684847107, −3.93548201746261355693540656446, −3.42925330752225835710064683024, −2.94044036637155068795967479207, −2.91641057587129766210279750477, −2.08031041651437184934195529047, −1.85624874936692437366585141253, −0.895490388010564683519205549610, −0.75787380941680745239523734767,
0.75787380941680745239523734767, 0.895490388010564683519205549610, 1.85624874936692437366585141253, 2.08031041651437184934195529047, 2.91641057587129766210279750477, 2.94044036637155068795967479207, 3.42925330752225835710064683024, 3.93548201746261355693540656446, 4.27401694018432444927684847107, 4.71530969459032516177177655947, 4.99035317855142227504260741965, 5.44178023611126331662578335000, 5.99996789858576194144210721437, 6.09055078613316102753393896897, 6.98598071186865353445104349553, 7.01647883510405504944266751933, 7.39285385258005308466497507102, 7.55158769064946380971256046966, 8.322044087526028216359009444055, 8.440136194256352642617783361755
