| L(s) = 1 | − 0.656·2-s + 2.67·3-s − 1.56·4-s − 1.55·5-s − 1.75·6-s + 2.17·7-s + 2.34·8-s + 4.17·9-s + 1.02·10-s + 0.877·11-s − 4.20·12-s + 13-s − 1.42·14-s − 4.17·15-s + 1.59·16-s + 4.00·17-s − 2.74·18-s − 0.601·19-s + 2.44·20-s + 5.82·21-s − 0.576·22-s + 7.74·23-s + 6.27·24-s − 2.57·25-s − 0.656·26-s + 3.14·27-s − 3.41·28-s + ⋯ |
| L(s) = 1 | − 0.464·2-s + 1.54·3-s − 0.784·4-s − 0.697·5-s − 0.718·6-s + 0.822·7-s + 0.828·8-s + 1.39·9-s + 0.323·10-s + 0.264·11-s − 1.21·12-s + 0.277·13-s − 0.382·14-s − 1.07·15-s + 0.399·16-s + 0.971·17-s − 0.646·18-s − 0.137·19-s + 0.546·20-s + 1.27·21-s − 0.122·22-s + 1.61·23-s + 1.28·24-s − 0.514·25-s − 0.128·26-s + 0.604·27-s − 0.645·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.535015697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.535015697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.656T + 2T^{2} \) |
| 3 | \( 1 - 2.67T + 3T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 - 0.877T + 11T^{2} \) |
| 17 | \( 1 - 4.00T + 17T^{2} \) |
| 19 | \( 1 + 0.601T + 19T^{2} \) |
| 23 | \( 1 - 7.74T + 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 37 | \( 1 + 2.60T + 37T^{2} \) |
| 41 | \( 1 + 5.00T + 41T^{2} \) |
| 43 | \( 1 + 2.25T + 43T^{2} \) |
| 47 | \( 1 + 1.99T + 47T^{2} \) |
| 53 | \( 1 + 3.22T + 53T^{2} \) |
| 59 | \( 1 - 6.18T + 59T^{2} \) |
| 61 | \( 1 - 2.46T + 61T^{2} \) |
| 67 | \( 1 + 6.59T + 67T^{2} \) |
| 71 | \( 1 + 0.570T + 71T^{2} \) |
| 73 | \( 1 + 6.17T + 73T^{2} \) |
| 79 | \( 1 - 9.85T + 79T^{2} \) |
| 83 | \( 1 - 7.25T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−11.10831147642673978429636224580, −10.03799722306106010357211264362, −9.196447252286113706924079638041, −8.398888122566616187779623380334, −8.021691756039554883601111958678, −7.11485570698157016764496732036, −5.13343666530974458056769437166, −4.09393056291002306566429244527, −3.17711986111718680614910518331, −1.43535114902336665308837251587,
1.43535114902336665308837251587, 3.17711986111718680614910518331, 4.09393056291002306566429244527, 5.13343666530974458056769437166, 7.11485570698157016764496732036, 8.021691756039554883601111958678, 8.398888122566616187779623380334, 9.196447252286113706924079638041, 10.03799722306106010357211264362, 11.10831147642673978429636224580
