| L(s) = 1 | − 2.56·2-s − 3-s + 4.56·4-s + 3.56·5-s + 2.56·6-s − 6.56·8-s + 9-s − 9.12·10-s + 1.56·11-s − 4.56·12-s + 0.438·13-s − 3.56·15-s + 7.68·16-s + 17-s − 2.56·18-s − 4.68·19-s + 16.2·20-s − 4·22-s − 2.43·23-s + 6.56·24-s + 7.68·25-s − 1.12·26-s − 27-s − 8.24·29-s + 9.12·30-s + 3.12·31-s − 6.56·32-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 0.577·3-s + 2.28·4-s + 1.59·5-s + 1.04·6-s − 2.31·8-s + 0.333·9-s − 2.88·10-s + 0.470·11-s − 1.31·12-s + 0.121·13-s − 0.919·15-s + 1.92·16-s + 0.242·17-s − 0.603·18-s − 1.07·19-s + 3.63·20-s − 0.852·22-s − 0.508·23-s + 1.33·24-s + 1.53·25-s − 0.220·26-s − 0.192·27-s − 1.53·29-s + 1.66·30-s + 0.560·31-s − 1.15·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4290364816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4290364816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 9.36T + 79T^{2} \) |
| 83 | \( 1 + 0.876T + 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−16.19067449482980868794416718938, −14.68468483701697200839391613200, −13.09137827239781845451040732831, −11.56366881196579548125954310900, −10.39218562652924539133521407122, −9.703880567543295830748797082163, −8.601123624966040242969084129386, −6.88591723922472899575565825980, −5.85807041859657629945277249827, −1.85697923917174874825727709922,
1.85697923917174874825727709922, 5.85807041859657629945277249827, 6.88591723922472899575565825980, 8.601123624966040242969084129386, 9.703880567543295830748797082163, 10.39218562652924539133521407122, 11.56366881196579548125954310900, 13.09137827239781845451040732831, 14.68468483701697200839391613200, 16.19067449482980868794416718938
