| L(s) = 1 | − 3-s + 2·5-s − 5·7-s − 9-s − 8·11-s − 13-s − 2·15-s + 3·17-s − 2·19-s + 5·21-s − 23-s + 3·25-s + 11·29-s + 2·31-s + 8·33-s − 10·35-s − 6·37-s + 39-s − 4·43-s − 2·45-s + 2·47-s + 9·49-s − 3·51-s − 9·53-s − 16·55-s + 2·57-s + 5·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.88·7-s − 1/3·9-s − 2.41·11-s − 0.277·13-s − 0.516·15-s + 0.727·17-s − 0.458·19-s + 1.09·21-s − 0.208·23-s + 3/5·25-s + 2.04·29-s + 0.359·31-s + 1.39·33-s − 1.69·35-s − 0.986·37-s + 0.160·39-s − 0.609·43-s − 0.298·45-s + 0.291·47-s + 9/7·49-s − 0.420·51-s − 1.23·53-s − 2.15·55-s + 0.264·57-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + T + 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 122 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 186 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 17 T + 202 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 60 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.964999336848016629304640193541, −7.41672307028850976195619430650, −7.13000328190260261705360641600, −6.70040856499583902766787921154, −6.37419950230333509976287115560, −6.23648531246965530345979061909, −5.58519760762751725717455767764, −5.55893235802298065784414345147, −5.09919841969148563171500095055, −4.91094277981190965344838062078, −4.27198913118696900252116090180, −3.79366608825540564603916404990, −3.13513070501732698910189526171, −3.03176645962550756098730208420, −2.50579502635435822342455277034, −2.43059702944974876373331108624, −1.58093590209432745577214821678, −0.873045285012107070038820027341, 0, 0,
0.873045285012107070038820027341, 1.58093590209432745577214821678, 2.43059702944974876373331108624, 2.50579502635435822342455277034, 3.03176645962550756098730208420, 3.13513070501732698910189526171, 3.79366608825540564603916404990, 4.27198913118696900252116090180, 4.91094277981190965344838062078, 5.09919841969148563171500095055, 5.55893235802298065784414345147, 5.58519760762751725717455767764, 6.23648531246965530345979061909, 6.37419950230333509976287115560, 6.70040856499583902766787921154, 7.13000328190260261705360641600, 7.41672307028850976195619430650, 7.964999336848016629304640193541
