| L(s) = 1 | − 4·7-s − 4·9-s − 4·11-s + 4·13-s + 2·19-s + 4·23-s − 4·29-s − 8·31-s − 4·37-s − 4·41-s − 12·43-s − 4·47-s + 6·49-s + 4·53-s + 8·59-s + 16·63-s + 8·73-s + 16·77-s + 7·81-s + 20·83-s − 4·89-s − 16·91-s + 12·97-s + 16·99-s − 16·101-s − 24·107-s − 4·109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 4/3·9-s − 1.20·11-s + 1.10·13-s + 0.458·19-s + 0.834·23-s − 0.742·29-s − 1.43·31-s − 0.657·37-s − 0.624·41-s − 1.82·43-s − 0.583·47-s + 6/7·49-s + 0.549·53-s + 1.04·59-s + 2.01·63-s + 0.936·73-s + 1.82·77-s + 7/9·81-s + 2.19·83-s − 0.423·89-s − 1.67·91-s + 1.21·97-s + 1.60·99-s − 1.59·101-s − 2.32·107-s − 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 234 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 180 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.79432407399651957656499311274, −7.26360037702473816286831300699, −6.85494484102937935225660226950, −6.81663171720674129539553253691, −6.17484621338640616532493773334, −6.09826479160781406640033671209, −5.46362728292603034051968087106, −5.37208300068153671445227872759, −5.10962786040283250059081653032, −4.61533135288556022239495548905, −3.75908366463655372624979375059, −3.67452503050343059814671555344, −3.30018720360330365272177445769, −3.08324764965632710496800547056, −2.52885265058266556684499462271, −2.16365361186484845449633454251, −1.56208210430134279045647431715, −0.863894482257697502357321524193, 0, 0,
0.863894482257697502357321524193, 1.56208210430134279045647431715, 2.16365361186484845449633454251, 2.52885265058266556684499462271, 3.08324764965632710496800547056, 3.30018720360330365272177445769, 3.67452503050343059814671555344, 3.75908366463655372624979375059, 4.61533135288556022239495548905, 5.10962786040283250059081653032, 5.37208300068153671445227872759, 5.46362728292603034051968087106, 6.09826479160781406640033671209, 6.17484621338640616532493773334, 6.81663171720674129539553253691, 6.85494484102937935225660226950, 7.26360037702473816286831300699, 7.79432407399651957656499311274
