| L(s) = 1 | + 3-s − 1.74·5-s − 2.35·7-s + 9-s − 4.17·11-s − 1.58·13-s − 1.74·15-s − 2.99·17-s − 6.10·19-s − 2.35·21-s + 2.00·23-s − 1.96·25-s + 27-s − 3.60·29-s − 3.07·31-s − 4.17·33-s + 4.10·35-s + 3.24·37-s − 1.58·39-s + 6.34·41-s + 1.84·43-s − 1.74·45-s − 2.38·47-s − 1.46·49-s − 2.99·51-s − 0.819·53-s + 7.28·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.779·5-s − 0.889·7-s + 0.333·9-s − 1.25·11-s − 0.438·13-s − 0.449·15-s − 0.726·17-s − 1.40·19-s − 0.513·21-s + 0.418·23-s − 0.392·25-s + 0.192·27-s − 0.668·29-s − 0.552·31-s − 0.727·33-s + 0.693·35-s + 0.532·37-s − 0.253·39-s + 0.990·41-s + 0.281·43-s − 0.259·45-s − 0.347·47-s − 0.208·49-s − 0.419·51-s − 0.112·53-s + 0.981·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7484539330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7484539330\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 1.74T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 + 4.17T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 + 2.99T + 17T^{2} \) |
| 19 | \( 1 + 6.10T + 19T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 + 3.60T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 - 3.24T + 37T^{2} \) |
| 41 | \( 1 - 6.34T + 41T^{2} \) |
| 43 | \( 1 - 1.84T + 43T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 + 0.819T + 53T^{2} \) |
| 59 | \( 1 - 6.19T + 59T^{2} \) |
| 61 | \( 1 - 8.06T + 61T^{2} \) |
| 67 | \( 1 - 1.44T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 0.554T + 83T^{2} \) |
| 89 | \( 1 + 7.50T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.73030634429700392481030139067, −7.35035943100511778990676198829, −6.55201103758603761915635735472, −5.82391193715948569455973904401, −4.87988571361853550025456677092, −4.17406589991441766069310565783, −3.51401817744886513017822581935, −2.65672118184334811355308183699, −2.08167324188837022699715080802, −0.38443283726831235033085892609,
0.38443283726831235033085892609, 2.08167324188837022699715080802, 2.65672118184334811355308183699, 3.51401817744886513017822581935, 4.17406589991441766069310565783, 4.87988571361853550025456677092, 5.82391193715948569455973904401, 6.55201103758603761915635735472, 7.35035943100511778990676198829, 7.73030634429700392481030139067
