| L(s) = 1 | − 2.46·3-s + 0.710·7-s + 3.09·9-s − 5.37·11-s − 6.61·13-s + 7.22·17-s + 2.14·19-s − 1.75·21-s − 3.32·23-s − 0.228·27-s − 3.56·29-s − 5.67·31-s + 13.2·33-s − 3.28·37-s + 16.3·39-s − 1.46·41-s − 0.681·43-s + 5.37·47-s − 6.49·49-s − 17.8·51-s + 0.624·53-s − 5.29·57-s − 9.90·59-s − 5.84·61-s + 2.19·63-s + 8.00·67-s + 8.20·69-s + ⋯ |
| L(s) = 1 | − 1.42·3-s + 0.268·7-s + 1.03·9-s − 1.62·11-s − 1.83·13-s + 1.75·17-s + 0.491·19-s − 0.382·21-s − 0.692·23-s − 0.0439·27-s − 0.661·29-s − 1.01·31-s + 2.30·33-s − 0.540·37-s + 2.61·39-s − 0.229·41-s − 0.103·43-s + 0.784·47-s − 0.927·49-s − 2.49·51-s + 0.0857·53-s − 0.700·57-s − 1.28·59-s − 0.748·61-s + 0.276·63-s + 0.977·67-s + 0.987·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3557179354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3557179354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 7 | \( 1 - 0.710T + 7T^{2} \) |
| 11 | \( 1 + 5.37T + 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 17 | \( 1 - 7.22T + 17T^{2} \) |
| 19 | \( 1 - 2.14T + 19T^{2} \) |
| 23 | \( 1 + 3.32T + 23T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 + 5.67T + 31T^{2} \) |
| 37 | \( 1 + 3.28T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 + 0.681T + 43T^{2} \) |
| 47 | \( 1 - 5.37T + 47T^{2} \) |
| 53 | \( 1 - 0.624T + 53T^{2} \) |
| 59 | \( 1 + 9.90T + 59T^{2} \) |
| 61 | \( 1 + 5.84T + 61T^{2} \) |
| 67 | \( 1 - 8.00T + 67T^{2} \) |
| 71 | \( 1 + 8.07T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 7.66T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 - 2.02T + 89T^{2} \) |
| 97 | \( 1 + 1.79T + 97T^{2} \) |
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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
−7.66538910162634740762252970905, −7.08282810488586041255445354001, −6.05647324097919181080388499488, −5.38914275619029221028807044852, −5.22478200411635939389542132526, −4.54435064241917839465199376527, −3.38130414162323857369364913148, −2.56823931943438473409443692750, −1.56189346755376379077280435996, −0.30165287238156929739854098448,
0.30165287238156929739854098448, 1.56189346755376379077280435996, 2.56823931943438473409443692750, 3.38130414162323857369364913148, 4.54435064241917839465199376527, 5.22478200411635939389542132526, 5.38914275619029221028807044852, 6.05647324097919181080388499488, 7.08282810488586041255445354001, 7.66538910162634740762252970905
