L(s) = 1 | + 0.965·3-s − 3.70·5-s + 7-s − 2.06·9-s − 1.96·11-s − 3.66·13-s − 3.57·15-s − 17-s − 0.875·19-s + 0.965·21-s − 1.69·23-s + 8.74·25-s − 4.89·27-s − 3.96·29-s − 3.85·31-s − 1.90·33-s − 3.70·35-s − 2.41·37-s − 3.53·39-s + 5.75·41-s + 1.27·43-s + 7.66·45-s − 6.74·47-s + 49-s − 0.965·51-s − 6.61·53-s + 7.30·55-s + ⋯ |
L(s) = 1 | + 0.557·3-s − 1.65·5-s + 0.377·7-s − 0.689·9-s − 0.593·11-s − 1.01·13-s − 0.923·15-s − 0.242·17-s − 0.200·19-s + 0.210·21-s − 0.353·23-s + 1.74·25-s − 0.941·27-s − 0.737·29-s − 0.692·31-s − 0.330·33-s − 0.626·35-s − 0.397·37-s − 0.566·39-s + 0.898·41-s + 0.193·43-s + 1.14·45-s − 0.983·47-s + 0.142·49-s − 0.135·51-s − 0.908·53-s + 0.984·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6486091807\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6486091807\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 - 0.965T + 3T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 + 3.66T + 13T^{2} \) |
| 19 | \( 1 + 0.875T + 19T^{2} \) |
| 23 | \( 1 + 1.69T + 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 2.41T + 37T^{2} \) |
| 41 | \( 1 - 5.75T + 41T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 - 1.02T + 59T^{2} \) |
| 61 | \( 1 - 1.56T + 61T^{2} \) |
| 67 | \( 1 - 1.34T + 67T^{2} \) |
| 71 | \( 1 + 2.42T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 - 3.53T + 79T^{2} \) |
| 83 | \( 1 + 7.24T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 3.45T + 97T^{2} \) |
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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
−8.062839207930666372758422113377, −7.41737737614603046843415924170, −6.75548067831594545279150191000, −5.62810101434310771241384475522, −4.95522587686765590853225532233, −4.20942967420999127609720125104, −3.53236712208068030971805258631, −2.80511285867930829210167008958, −1.98177799299688603017261543061, −0.36715323660483673765860538948,
0.36715323660483673765860538948, 1.98177799299688603017261543061, 2.80511285867930829210167008958, 3.53236712208068030971805258631, 4.20942967420999127609720125104, 4.95522587686765590853225532233, 5.62810101434310771241384475522, 6.75548067831594545279150191000, 7.41737737614603046843415924170, 8.062839207930666372758422113377