| L(s) = 1 | + 4.04·2-s − 6.52·3-s + 8.39·4-s − 26.4·6-s − 7·7-s + 1.58·8-s + 15.6·9-s + 6.78·11-s − 54.7·12-s − 48.9·13-s − 28.3·14-s − 60.7·16-s − 92.4·17-s + 63.1·18-s − 125.·19-s + 45.6·21-s + 27.4·22-s + 32.2·23-s − 10.3·24-s − 198.·26-s + 74.3·27-s − 58.7·28-s + 282.·29-s + 205.·31-s − 258.·32-s − 44.2·33-s − 374.·34-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 1.25·3-s + 1.04·4-s − 1.79·6-s − 0.377·7-s + 0.0698·8-s + 0.578·9-s + 0.185·11-s − 1.31·12-s − 1.04·13-s − 0.541·14-s − 0.948·16-s − 1.31·17-s + 0.827·18-s − 1.51·19-s + 0.474·21-s + 0.266·22-s + 0.292·23-s − 0.0877·24-s − 1.49·26-s + 0.530·27-s − 0.396·28-s + 1.81·29-s + 1.19·31-s − 1.42·32-s − 0.233·33-s − 1.88·34-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.04T + 8T^{2} \) |
| 3 | \( 1 + 6.52T + 27T^{2} \) |
| 11 | \( 1 - 6.78T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 92.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 32.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 282.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 205.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 123.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 35.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 419.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 0.365T + 1.48e5T^{2} \) |
| 59 | \( 1 - 328.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 515.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 828.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 496.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 701.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 199.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 194.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 137.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 220.T + 9.12e5T^{2} \) |
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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
−12.03800311785481645089385689920, −11.16346308153900662304801112102, −10.18794925764385988798726158836, −8.692080342934056033971275683084, −6.70309320644208085946320962263, −6.34139052150607437890645164701, −5.01780550110484013517578075560, −4.37161285307502519328235807274, −2.63054893822071619729802472483, 0,
2.63054893822071619729802472483, 4.37161285307502519328235807274, 5.01780550110484013517578075560, 6.34139052150607437890645164701, 6.70309320644208085946320962263, 8.692080342934056033971275683084, 10.18794925764385988798726158836, 11.16346308153900662304801112102, 12.03800311785481645089385689920
