| L(s) = 1 | + 4.87·2-s + 4.14·3-s + 15.7·4-s + 20.2·6-s − 7·7-s + 37.6·8-s − 9.78·9-s + 36.9·11-s + 65.2·12-s + 61.3·13-s − 34.1·14-s + 57.7·16-s + 44.8·17-s − 47.6·18-s − 139.·19-s − 29.0·21-s + 180.·22-s − 217.·23-s + 156.·24-s + 298.·26-s − 152.·27-s − 110.·28-s − 33.8·29-s + 124.·31-s − 20.2·32-s + 153.·33-s + 218.·34-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 0.798·3-s + 1.96·4-s + 1.37·6-s − 0.377·7-s + 1.66·8-s − 0.362·9-s + 1.01·11-s + 1.57·12-s + 1.30·13-s − 0.651·14-s + 0.902·16-s + 0.639·17-s − 0.624·18-s − 1.68·19-s − 0.301·21-s + 1.74·22-s − 1.97·23-s + 1.33·24-s + 2.25·26-s − 1.08·27-s − 0.743·28-s − 0.216·29-s + 0.720·31-s − 0.111·32-s + 0.809·33-s + 1.10·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.426929207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.426929207\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 4.87T + 8T^{2} \) |
| 3 | \( 1 - 4.14T + 27T^{2} \) |
| 11 | \( 1 - 36.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 61.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 217.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 33.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 237.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 343.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 16.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 346.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 135.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 490.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 477.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 45.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 100.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 880.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 619.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 231.T + 9.12e5T^{2} \) |
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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
−12.47442545267096505965635818931, −11.67302735266594016745742458167, −10.59589615465738106769034969041, −9.096500229355524334596048729139, −8.043101483122425136014638530171, −6.45435649022378330035426048120, −5.89156880718892164262276887496, −4.13589209371344773455849592961, −3.50956635581667309636234536906, −2.11137830062807610517246830535,
2.11137830062807610517246830535, 3.50956635581667309636234536906, 4.13589209371344773455849592961, 5.89156880718892164262276887496, 6.45435649022378330035426048120, 8.043101483122425136014638530171, 9.096500229355524334596048729139, 10.59589615465738106769034969041, 11.67302735266594016745742458167, 12.47442545267096505965635818931
