L(s) = 1 | − 1.01·3-s − 1.29·5-s − 7-s − 1.96·9-s − 3.99·11-s − 0.534·13-s + 1.31·15-s − 6.47·17-s + 3.19·19-s + 1.01·21-s − 2.28·23-s − 3.33·25-s + 5.04·27-s + 6.09·29-s − 5.70·31-s + 4.05·33-s + 1.29·35-s − 6.44·37-s + 0.543·39-s − 4.27·41-s + 5.11·43-s + 2.53·45-s − 7.83·47-s + 49-s + 6.58·51-s − 13.5·53-s + 5.15·55-s + ⋯ |
L(s) = 1 | − 0.587·3-s − 0.577·5-s − 0.377·7-s − 0.655·9-s − 1.20·11-s − 0.148·13-s + 0.339·15-s − 1.57·17-s + 0.731·19-s + 0.221·21-s − 0.476·23-s − 0.666·25-s + 0.971·27-s + 1.13·29-s − 1.02·31-s + 0.706·33-s + 0.218·35-s − 1.05·37-s + 0.0871·39-s − 0.668·41-s + 0.780·43-s + 0.378·45-s − 1.14·47-s + 0.142·49-s + 0.921·51-s − 1.86·53-s + 0.695·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4016865422\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4016865422\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 1.01T + 3T^{2} \) |
| 5 | \( 1 + 1.29T + 5T^{2} \) |
| 11 | \( 1 + 3.99T + 11T^{2} \) |
| 13 | \( 1 + 0.534T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 + 2.28T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 + 6.44T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 - 5.11T + 43T^{2} \) |
| 47 | \( 1 + 7.83T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 9.71T + 59T^{2} \) |
| 61 | \( 1 - 2.06T + 61T^{2} \) |
| 67 | \( 1 - 9.06T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 8.18T + 73T^{2} \) |
| 79 | \( 1 + 0.960T + 79T^{2} \) |
| 83 | \( 1 + 5.40T + 83T^{2} \) |
| 89 | \( 1 - 4.97T + 89T^{2} \) |
| 97 | \( 1 - 6.97T + 97T^{2} \) |
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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
−8.354435224523887897554861588790, −7.961817670931004316781662686872, −6.93693993891297340622464716489, −6.40756044415945095756885945494, −5.39709726271998272962400903085, −4.96862568933439868052427551131, −3.90895048628221744986981041789, −3.01293808589605002116802251126, −2.11271482955653833086562327091, −0.35692526918097609459755742246,
0.35692526918097609459755742246, 2.11271482955653833086562327091, 3.01293808589605002116802251126, 3.90895048628221744986981041789, 4.96862568933439868052427551131, 5.39709726271998272962400903085, 6.40756044415945095756885945494, 6.93693993891297340622464716489, 7.961817670931004316781662686872, 8.354435224523887897554861588790