| L(s) = 1 | − 2-s + 3-s + 6·5-s − 6-s − 7-s + 8-s − 6·10-s + 11-s + 5·13-s + 14-s + 6·15-s − 16-s − 7·17-s + 6·19-s − 21-s − 22-s + 4·23-s + 24-s + 17·25-s − 5·26-s − 27-s − 10·29-s − 6·30-s + 33-s + 7·34-s − 6·35-s + 6·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 2.68·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s − 1.89·10-s + 0.301·11-s + 1.38·13-s + 0.267·14-s + 1.54·15-s − 1/4·16-s − 1.69·17-s + 1.37·19-s − 0.218·21-s − 0.213·22-s + 0.834·23-s + 0.204·24-s + 17/5·25-s − 0.980·26-s − 0.192·27-s − 1.85·29-s − 1.09·30-s + 0.174·33-s + 1.20·34-s − 1.01·35-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736164 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736164 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.988012750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.988012750\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−10.01507939866371235524724062637, −9.974444819978876558871335817494, −9.441089292581048816776607304451, −9.088578355643271580488207002825, −8.987887246682937807339299309871, −8.578478080221084244563815426833, −8.016935954366676584265980099733, −7.36212762031649593750468966401, −6.78580949089111307768527417955, −6.65371461092176648567221449283, −5.90633782621818883747887856946, −5.80184926444342598251690473090, −5.23092137460280078003826902796, −4.76962240806989803867262623103, −3.72490513196208490160295946702, −3.58105115295431297537451284172, −2.49153018513073731203798188425, −2.27748880784208577149713145657, −1.62977127396361109794241611660, −0.991615574428848643685558750954,
0.991615574428848643685558750954, 1.62977127396361109794241611660, 2.27748880784208577149713145657, 2.49153018513073731203798188425, 3.58105115295431297537451284172, 3.72490513196208490160295946702, 4.76962240806989803867262623103, 5.23092137460280078003826902796, 5.80184926444342598251690473090, 5.90633782621818883747887856946, 6.65371461092176648567221449283, 6.78580949089111307768527417955, 7.36212762031649593750468966401, 8.016935954366676584265980099733, 8.578478080221084244563815426833, 8.987887246682937807339299309871, 9.088578355643271580488207002825, 9.441089292581048816776607304451, 9.974444819978876558871335817494, 10.01507939866371235524724062637
