| L(s) = 1 | + 2-s + 3-s + 4-s − 0.555·5-s + 6-s − 7-s + 8-s + 9-s − 0.555·10-s + 1.39·11-s + 12-s − 13-s − 14-s − 0.555·15-s + 16-s + 17-s + 18-s − 8.27·19-s − 0.555·20-s − 21-s + 1.39·22-s + 3.50·23-s + 24-s − 4.69·25-s − 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.248·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.175·10-s + 0.421·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.143·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 1.89·19-s − 0.124·20-s − 0.218·21-s + 0.297·22-s + 0.730·23-s + 0.204·24-s − 0.938·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.839924955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.839924955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 0.555T + 5T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 19 | \( 1 + 8.27T + 19T^{2} \) |
| 23 | \( 1 - 3.50T + 23T^{2} \) |
| 29 | \( 1 - 8.96T + 29T^{2} \) |
| 31 | \( 1 - 5.16T + 31T^{2} \) |
| 37 | \( 1 + 8.03T + 37T^{2} \) |
| 41 | \( 1 - 1.96T + 41T^{2} \) |
| 43 | \( 1 - 8.77T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 7.97T + 53T^{2} \) |
| 59 | \( 1 + 2.01T + 59T^{2} \) |
| 61 | \( 1 + 8.17T + 61T^{2} \) |
| 67 | \( 1 - 8.56T + 67T^{2} \) |
| 71 | \( 1 - 9.45T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 + 4.13T + 79T^{2} \) |
| 83 | \( 1 - 2.72T + 83T^{2} \) |
| 89 | \( 1 + 3.42T + 89T^{2} \) |
| 97 | \( 1 - 4.98T + 97T^{2} \) |
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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
−7.57195201991952171863188711423, −6.98473138148872408192258120495, −6.34167106563309907674526077009, −5.74018411694732542110293831760, −4.66322939929717697982666704313, −4.23563727408429816207812111610, −3.51812854638226462628540571855, −2.67891961556990087242122756142, −2.07275062767436124787152319473, −0.812886508339545283011530035779,
0.812886508339545283011530035779, 2.07275062767436124787152319473, 2.67891961556990087242122756142, 3.51812854638226462628540571855, 4.23563727408429816207812111610, 4.66322939929717697982666704313, 5.74018411694732542110293831760, 6.34167106563309907674526077009, 6.98473138148872408192258120495, 7.57195201991952171863188711423
