| L(s) = 1 | − 0.827·2-s − 1.74·3-s − 1.31·4-s − 0.781·5-s + 1.44·6-s − 7-s + 2.74·8-s + 0.0381·9-s + 0.646·10-s − 0.696·11-s + 2.29·12-s + 0.888·13-s + 0.827·14-s + 1.36·15-s + 0.361·16-s + 1.72·17-s − 0.0315·18-s + 7.17·19-s + 1.02·20-s + 1.74·21-s + 0.576·22-s + 5.79·23-s − 4.78·24-s − 4.38·25-s − 0.734·26-s + 5.16·27-s + 1.31·28-s + ⋯ |
| L(s) = 1 | − 0.585·2-s − 1.00·3-s − 0.657·4-s − 0.349·5-s + 0.588·6-s − 0.377·7-s + 0.969·8-s + 0.0127·9-s + 0.204·10-s − 0.210·11-s + 0.661·12-s + 0.246·13-s + 0.221·14-s + 0.351·15-s + 0.0904·16-s + 0.417·17-s − 0.00744·18-s + 1.64·19-s + 0.229·20-s + 0.380·21-s + 0.122·22-s + 1.20·23-s − 0.975·24-s − 0.877·25-s − 0.144·26-s + 0.993·27-s + 0.248·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4743118824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4743118824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.827T + 2T^{2} \) |
| 3 | \( 1 + 1.74T + 3T^{2} \) |
| 5 | \( 1 + 0.781T + 5T^{2} \) |
| 11 | \( 1 + 0.696T + 11T^{2} \) |
| 13 | \( 1 - 0.888T + 13T^{2} \) |
| 17 | \( 1 - 1.72T + 17T^{2} \) |
| 19 | \( 1 - 7.17T + 19T^{2} \) |
| 23 | \( 1 - 5.79T + 23T^{2} \) |
| 29 | \( 1 - 4.84T + 29T^{2} \) |
| 31 | \( 1 - 2.61T + 31T^{2} \) |
| 37 | \( 1 + 8.37T + 37T^{2} \) |
| 41 | \( 1 + 0.0770T + 41T^{2} \) |
| 47 | \( 1 - 3.18T + 47T^{2} \) |
| 53 | \( 1 + 0.507T + 53T^{2} \) |
| 59 | \( 1 - 4.48T + 59T^{2} \) |
| 61 | \( 1 + 1.99T + 61T^{2} \) |
| 67 | \( 1 + 4.72T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 + 5.79T + 79T^{2} \) |
| 83 | \( 1 - 4.18T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−11.67492923798253841141126134054, −10.72103618816464709119309743244, −9.923537169394974419362940419033, −8.980904277034312799873995636960, −7.967725942081695075669341067008, −6.95310542149712637889733982216, −5.63487914653584810425690922187, −4.84745373047723984926044410847, −3.37446497723967876494654150815, −0.823724309778016299604679645331,
0.823724309778016299604679645331, 3.37446497723967876494654150815, 4.84745373047723984926044410847, 5.63487914653584810425690922187, 6.95310542149712637889733982216, 7.967725942081695075669341067008, 8.980904277034312799873995636960, 9.923537169394974419362940419033, 10.72103618816464709119309743244, 11.67492923798253841141126134054
