| L(s) = 1 | + 2·3-s + 4·7-s + 3·9-s + 8·21-s + 8·23-s + 4·27-s + 8·29-s + 20·41-s − 8·43-s + 16·47-s − 2·49-s + 20·61-s + 12·63-s − 16·67-s + 16·69-s + 5·81-s − 8·83-s + 16·87-s + 12·89-s + 24·101-s − 28·103-s − 24·107-s + 20·109-s − 2·121-s + 40·123-s + 127-s − 16·129-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 9-s + 1.74·21-s + 1.66·23-s + 0.769·27-s + 1.48·29-s + 3.12·41-s − 1.21·43-s + 2.33·47-s − 2/7·49-s + 2.56·61-s + 1.51·63-s − 1.95·67-s + 1.92·69-s + 5/9·81-s − 0.878·83-s + 1.71·87-s + 1.27·89-s + 2.38·101-s − 2.75·103-s − 2.32·107-s + 1.91·109-s − 0.181·121-s + 3.60·123-s + 0.0887·127-s − 1.40·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.478034352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.478034352\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
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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
−8.989687031930457135999193158593, −8.828228639670930280770495358102, −8.377646859695115654245638810930, −8.050894721563563897044808796100, −7.66209956965641345430393745894, −7.45555910039859197941331685563, −6.88689601251649846872269261400, −6.69581814110370211988702923670, −5.97026279383752351353144480738, −5.62347779434008909036818340706, −4.98876645031598159722284006617, −4.81875327190616948962911925824, −4.24537533360346946289565811614, −4.06954958122591432742831016304, −3.31391313498403849394179430161, −2.87151009502897082228150529747, −2.44515175279066703071556712507, −2.02305164280206844387830562801, −1.17476725028721929729515509850, −0.969589636468131946357116609884,
0.969589636468131946357116609884, 1.17476725028721929729515509850, 2.02305164280206844387830562801, 2.44515175279066703071556712507, 2.87151009502897082228150529747, 3.31391313498403849394179430161, 4.06954958122591432742831016304, 4.24537533360346946289565811614, 4.81875327190616948962911925824, 4.98876645031598159722284006617, 5.62347779434008909036818340706, 5.97026279383752351353144480738, 6.69581814110370211988702923670, 6.88689601251649846872269261400, 7.45555910039859197941331685563, 7.66209956965641345430393745894, 8.050894721563563897044808796100, 8.377646859695115654245638810930, 8.828228639670930280770495358102, 8.989687031930457135999193158593
