| L(s) = 1 | + 3.11·3-s + 3.12·5-s − 1.63·7-s + 6.67·9-s − 1.72·11-s − 1.82·13-s + 9.70·15-s + 17-s + 4.18·19-s − 5.08·21-s + 5.92·23-s + 4.73·25-s + 11.4·27-s − 0.256·29-s + 0.752·31-s − 5.35·33-s − 5.09·35-s + 2.71·37-s − 5.66·39-s + 1.61·41-s + 1.84·43-s + 20.8·45-s − 11.3·47-s − 4.32·49-s + 3.11·51-s + 8.07·53-s − 5.37·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s + 1.39·5-s − 0.617·7-s + 2.22·9-s − 0.519·11-s − 0.505·13-s + 2.50·15-s + 0.242·17-s + 0.960·19-s − 1.10·21-s + 1.23·23-s + 0.947·25-s + 2.20·27-s − 0.0476·29-s + 0.135·31-s − 0.932·33-s − 0.861·35-s + 0.446·37-s − 0.907·39-s + 0.251·41-s + 0.281·43-s + 3.10·45-s − 1.66·47-s − 0.618·49-s + 0.435·51-s + 1.10·53-s − 0.724·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.754534614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.754534614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 3.11T + 3T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 + 1.72T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 19 | \( 1 - 4.18T + 19T^{2} \) |
| 23 | \( 1 - 5.92T + 23T^{2} \) |
| 29 | \( 1 + 0.256T + 29T^{2} \) |
| 31 | \( 1 - 0.752T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 - 1.61T + 41T^{2} \) |
| 43 | \( 1 - 1.84T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 8.07T + 53T^{2} \) |
| 61 | \( 1 + 6.80T + 61T^{2} \) |
| 67 | \( 1 - 4.90T + 67T^{2} \) |
| 71 | \( 1 + 3.75T + 71T^{2} \) |
| 73 | \( 1 + 8.68T + 73T^{2} \) |
| 79 | \( 1 - 3.18T + 79T^{2} \) |
| 83 | \( 1 - 1.57T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 5.66T + 97T^{2} \) |
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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
−8.593624691735231101860497344347, −7.74954731545391575325195774567, −7.16133050001806627723210386194, −6.37448706715313489178777698809, −5.41693555996391054246521442258, −4.65209233252579417139117110115, −3.42788347008332444792462446919, −2.88089128608927805508909952707, −2.23531720887167238701943688379, −1.27256309542509453170597564352,
1.27256309542509453170597564352, 2.23531720887167238701943688379, 2.88089128608927805508909952707, 3.42788347008332444792462446919, 4.65209233252579417139117110115, 5.41693555996391054246521442258, 6.37448706715313489178777698809, 7.16133050001806627723210386194, 7.74954731545391575325195774567, 8.593624691735231101860497344347
