L(s) = 1 | + 2·3-s + 3·9-s − 4·13-s − 14·17-s − 10·23-s − 25-s + 4·27-s − 20·29-s − 8·39-s + 20·43-s − 11·49-s − 28·51-s + 6·53-s − 6·61-s − 20·69-s − 2·75-s − 22·79-s + 5·81-s − 40·87-s − 20·101-s + 32·103-s − 6·107-s + 12·113-s − 12·117-s + 13·121-s + 127-s + 40·129-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 1.10·13-s − 3.39·17-s − 2.08·23-s − 1/5·25-s + 0.769·27-s − 3.71·29-s − 1.28·39-s + 3.04·43-s − 1.57·49-s − 3.92·51-s + 0.824·53-s − 0.768·61-s − 2.40·69-s − 0.230·75-s − 2.47·79-s + 5/9·81-s − 4.28·87-s − 1.99·101-s + 3.15·103-s − 0.580·107-s + 1.12·113-s − 1.10·117-s + 1.18·121-s + 0.0887·127-s + 3.52·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 129 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
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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
−8.651599867903158058579235104206, −8.069527235610206721637545731093, −7.71543960485720951215901632825, −7.51828269242858243395883582177, −6.97375707545871503590386797504, −6.95191434505101371282007176887, −6.16068857222911978553679690607, −5.90266472969227229826531677338, −5.58381022011482399534427468669, −4.79556756467795226289226496295, −4.46955196212405363697066272667, −4.15743339213029543714365971003, −3.85605536929498218460351649100, −3.37014939302658015710644685594, −2.54107756946499549681844909465, −2.26724809995220455000375157064, −2.09852142466896266384586799509, −1.53461649387954478402058724006, 0, 0,
1.53461649387954478402058724006, 2.09852142466896266384586799509, 2.26724809995220455000375157064, 2.54107756946499549681844909465, 3.37014939302658015710644685594, 3.85605536929498218460351649100, 4.15743339213029543714365971003, 4.46955196212405363697066272667, 4.79556756467795226289226496295, 5.58381022011482399534427468669, 5.90266472969227229826531677338, 6.16068857222911978553679690607, 6.95191434505101371282007176887, 6.97375707545871503590386797504, 7.51828269242858243395883582177, 7.71543960485720951215901632825, 8.069527235610206721637545731093, 8.651599867903158058579235104206