| L(s) = 1 | − 1.29·2-s + 2.84·3-s − 0.325·4-s − 3.15·5-s − 3.68·6-s − 5.03·7-s + 3.00·8-s + 5.12·9-s + 4.07·10-s − 0.928·12-s − 5.37·13-s + 6.50·14-s − 8.98·15-s − 3.24·16-s − 3.74·17-s − 6.62·18-s − 4.04·19-s + 1.02·20-s − 14.3·21-s − 8.31·23-s + 8.57·24-s + 4.94·25-s + 6.96·26-s + 6.04·27-s + 1.63·28-s + 2.39·29-s + 11.6·30-s + ⋯ |
| L(s) = 1 | − 0.914·2-s + 1.64·3-s − 0.162·4-s − 1.41·5-s − 1.50·6-s − 1.90·7-s + 1.06·8-s + 1.70·9-s + 1.29·10-s − 0.268·12-s − 1.49·13-s + 1.73·14-s − 2.32·15-s − 0.810·16-s − 0.907·17-s − 1.56·18-s − 0.927·19-s + 0.229·20-s − 3.12·21-s − 1.73·23-s + 1.75·24-s + 0.988·25-s + 1.36·26-s + 1.16·27-s + 0.309·28-s + 0.444·29-s + 2.12·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02533898109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02533898109\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 3 | \( 1 - 2.84T + 3T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 5.03T + 7T^{2} \) |
| 13 | \( 1 + 5.37T + 13T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 + 8.31T + 23T^{2} \) |
| 29 | \( 1 - 2.39T + 29T^{2} \) |
| 31 | \( 1 + 7.43T + 31T^{2} \) |
| 37 | \( 1 + 1.77T + 37T^{2} \) |
| 41 | \( 1 + 0.711T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 + 5.60T + 53T^{2} \) |
| 59 | \( 1 + 7.84T + 59T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 5.49T + 71T^{2} \) |
| 73 | \( 1 + 2.39T + 73T^{2} \) |
| 79 | \( 1 + 0.369T + 79T^{2} \) |
| 83 | \( 1 + 9.97T + 83T^{2} \) |
| 89 | \( 1 - 7.27T + 89T^{2} \) |
| 97 | \( 1 + 2.78T + 97T^{2} \) |
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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
−7.916837314561590865624295291533, −7.47882051910697278340167179128, −7.05147664549297855894152178143, −6.13006989502274917121232741403, −4.57943963886107909340763278665, −4.11009713112545784431425390226, −3.50500245040909207737675185281, −2.71644730164380419042105603920, −1.94733297735796746198303176842, −0.07869487437397386843477071061,
0.07869487437397386843477071061, 1.94733297735796746198303176842, 2.71644730164380419042105603920, 3.50500245040909207737675185281, 4.11009713112545784431425390226, 4.57943963886107909340763278665, 6.13006989502274917121232741403, 7.05147664549297855894152178143, 7.47882051910697278340167179128, 7.916837314561590865624295291533
