L(s) = 1 | + 3.93·2-s + 27·3-s − 112.·4-s − 305.·5-s + 106.·6-s − 1.40e3·7-s − 946.·8-s + 729·9-s − 1.20e3·10-s + 322.·11-s − 3.03e3·12-s − 6.38e3·13-s − 5.53e3·14-s − 8.25e3·15-s + 1.06e4·16-s − 1.48e4·17-s + 2.86e3·18-s − 1.27e4·19-s + 3.43e4·20-s − 3.79e4·21-s + 1.26e3·22-s − 1.73e4·23-s − 2.55e4·24-s + 1.52e4·25-s − 2.51e4·26-s + 1.96e4·27-s + 1.58e5·28-s + ⋯ |
L(s) = 1 | + 0.347·2-s + 0.577·3-s − 0.879·4-s − 1.09·5-s + 0.200·6-s − 1.55·7-s − 0.653·8-s + 0.333·9-s − 0.380·10-s + 0.0729·11-s − 0.507·12-s − 0.806·13-s − 0.538·14-s − 0.631·15-s + 0.652·16-s − 0.733·17-s + 0.115·18-s − 0.425·19-s + 0.961·20-s − 0.895·21-s + 0.0253·22-s − 0.296·23-s − 0.377·24-s + 0.195·25-s − 0.280·26-s + 0.192·27-s + 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.6376650875\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6376650875\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 27T \) |
| 59 | \( 1 - 2.05e5T \) |
good | 2 | \( 1 - 3.93T + 128T^{2} \) |
| 5 | \( 1 + 305.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.40e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 322.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.38e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.48e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.27e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.73e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 6.86e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.53e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.98e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.19e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.04e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.38e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.48e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 9.15e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.28e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.02e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.36e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.08e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.11e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.70e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.80e6T + 8.07e13T^{2} \) |
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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
−11.65930120914800275223122607731, −10.10300313788074030981995243751, −9.361427966614332037632600092469, −8.421096388297436496632737002672, −7.33201532981638774770527343924, −6.14236792989924801278744334186, −4.52441909139769438626611306879, −3.74747054919846346920659487175, −2.74429771129966258661547282486, −0.38451077005128205142020335772,
0.38451077005128205142020335772, 2.74429771129966258661547282486, 3.74747054919846346920659487175, 4.52441909139769438626611306879, 6.14236792989924801278744334186, 7.33201532981638774770527343924, 8.421096388297436496632737002672, 9.361427966614332037632600092469, 10.10300313788074030981995243751, 11.65930120914800275223122607731