L(s) = 1 | − 0.185·3-s − 3.29·5-s − 2.67·7-s − 2.96·9-s − 3.10·11-s − 5.10·13-s + 0.612·15-s + 1.75·17-s − 3.96·19-s + 0.496·21-s − 1.23·23-s + 5.85·25-s + 1.10·27-s − 2.68·29-s − 0.123·31-s + 0.577·33-s + 8.79·35-s − 7.07·37-s + 0.948·39-s − 4.20·41-s − 43-s + 9.77·45-s + 1.14·47-s + 0.131·49-s − 0.325·51-s + 11.6·53-s + 10.2·55-s + ⋯ |
L(s) = 1 | − 0.107·3-s − 1.47·5-s − 1.00·7-s − 0.988·9-s − 0.936·11-s − 1.41·13-s + 0.158·15-s + 0.424·17-s − 0.909·19-s + 0.108·21-s − 0.257·23-s + 1.17·25-s + 0.213·27-s − 0.498·29-s − 0.0221·31-s + 0.100·33-s + 1.48·35-s − 1.16·37-s + 0.151·39-s − 0.656·41-s − 0.152·43-s + 1.45·45-s + 0.166·47-s + 0.0188·49-s − 0.0456·51-s + 1.60·53-s + 1.37·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1142147827\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1142147827\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 0.185T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 11 | \( 1 + 3.10T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 19 | \( 1 + 3.96T + 19T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 + 2.68T + 29T^{2} \) |
| 31 | \( 1 + 0.123T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 47 | \( 1 - 1.14T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 0.143T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 7.63T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 1.16T + 89T^{2} \) |
| 97 | \( 1 + 7.53T + 97T^{2} \) |
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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
−8.633140777775052025262533083774, −8.062766318848821225995115222625, −7.34521639217847201706228696789, −6.70933630636398340907198268747, −5.64227831834301405201736546438, −4.94245431886113051775949584363, −3.93268563759966208375648187189, −3.16846246315625988527506676078, −2.40002615500226152100262098378, −0.19249320238573546797170423513,
0.19249320238573546797170423513, 2.40002615500226152100262098378, 3.16846246315625988527506676078, 3.93268563759966208375648187189, 4.94245431886113051775949584363, 5.64227831834301405201736546438, 6.70933630636398340907198268747, 7.34521639217847201706228696789, 8.062766318848821225995115222625, 8.633140777775052025262533083774