| L(s) = 1 | + 7·2-s − 15·4-s − 553·8-s + 1.08e3·13-s − 2.91e3·16-s + 1.21e4·23-s + 1.56e4·25-s + 7.57e3·26-s − 3.07e4·29-s + 5.87e4·31-s + 1.50e4·32-s − 4.36e4·41-s + 8.51e4·46-s + 2.05e5·47-s + 1.17e5·49-s + 1.09e5·50-s − 1.62e4·52-s − 2.15e5·58-s + 2.53e5·59-s + 4.11e5·62-s + 2.91e5·64-s − 6.67e5·71-s + 7.25e5·73-s − 3.05e5·82-s − 1.82e5·92-s + 1.43e6·94-s + 8.23e5·98-s + ⋯ |
| L(s) = 1 | + 7/8·2-s − 0.234·4-s − 1.08·8-s + 0.492·13-s − 0.710·16-s + 23-s + 25-s + 0.430·26-s − 1.26·29-s + 1.97·31-s + 0.458·32-s − 0.633·41-s + 7/8·46-s + 1.97·47-s + 49-s + 7/8·50-s − 0.115·52-s − 1.10·58-s + 1.23·59-s + 1.72·62-s + 1.11·64-s − 1.86·71-s + 1.86·73-s − 0.553·82-s − 0.234·92-s + 1.73·94-s + 7/8·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.691305730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.691305730\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - p^{3} T \) |
| good | 2 | \( 1 - 7 T + p^{6} T^{2} \) |
| 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 7 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( 1 - 1082 T + p^{6} T^{2} \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( 1 + 30746 T + p^{6} T^{2} \) |
| 31 | \( 1 - 58754 T + p^{6} T^{2} \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( 1 + 43634 T + p^{6} T^{2} \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( 1 - 205342 T + p^{6} T^{2} \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( 1 - 253942 T + p^{6} T^{2} \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( 1 + 667154 T + p^{6} T^{2} \) |
| 73 | \( 1 - 725042 T + p^{6} T^{2} \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.54716265197296038714606853706, −10.45554556745774000630636596661, −9.237354197399198739775347048622, −8.452792376307919635516206183209, −7.01084387251819033234980784863, −5.89538152077761041911763273143, −4.90776775002447377488476556734, −3.85443700897885935858428715053, −2.71077784958008177488851218020, −0.831458597353747937277135148967,
0.831458597353747937277135148967, 2.71077784958008177488851218020, 3.85443700897885935858428715053, 4.90776775002447377488476556734, 5.89538152077761041911763273143, 7.01084387251819033234980784863, 8.452792376307919635516206183209, 9.237354197399198739775347048622, 10.45554556745774000630636596661, 11.54716265197296038714606853706
