| L(s) = 1 | + 39.7·2-s + 81·3-s + 1.07e3·4-s − 1.73e3·5-s + 3.22e3·6-s − 3.88e3·7-s + 2.21e4·8-s + 6.56e3·9-s − 6.88e4·10-s + 5.68e4·11-s + 8.66e4·12-s − 1.09e5·13-s − 1.54e5·14-s − 1.40e5·15-s + 3.35e5·16-s − 6.53e5·17-s + 2.60e5·18-s − 1.75e5·19-s − 1.85e6·20-s − 3.14e5·21-s + 2.26e6·22-s − 1.42e6·23-s + 1.79e6·24-s + 1.04e6·25-s − 4.36e6·26-s + 5.31e5·27-s − 4.15e6·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 0.577·3-s + 2.09·4-s − 1.23·5-s + 1.01·6-s − 0.611·7-s + 1.91·8-s + 0.333·9-s − 2.17·10-s + 1.17·11-s + 1.20·12-s − 1.06·13-s − 1.07·14-s − 0.715·15-s + 1.27·16-s − 1.89·17-s + 0.585·18-s − 0.309·19-s − 2.58·20-s − 0.353·21-s + 2.05·22-s − 1.06·23-s + 1.10·24-s + 0.534·25-s − 1.87·26-s + 0.192·27-s − 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 - 39.7T + 512T^{2} \) |
| 5 | \( 1 + 1.73e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.88e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.68e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.09e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.53e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.75e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.42e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.18e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.64e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.37e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.93e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.28e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.30e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.99e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 1.37e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.40e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.40e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.45e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.07e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.98e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.64e6T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.02e9T + 7.60e17T^{2} \) |
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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
−11.06205568712215941971318410815, −9.541356896743079549455165749439, −8.221595392311813673516888338585, −6.97660867124570417629521762452, −6.39791647459469847907086429452, −4.63223373410960529915606151524, −4.10365915321777699805605431487, −3.16140777716975577044587829834, −2.06590637559572616597988737958, 0,
2.06590637559572616597988737958, 3.16140777716975577044587829834, 4.10365915321777699805605431487, 4.63223373410960529915606151524, 6.39791647459469847907086429452, 6.97660867124570417629521762452, 8.221595392311813673516888338585, 9.541356896743079549455165749439, 11.06205568712215941971318410815
