L(s) = 1 | + 2-s + 1.43·3-s + 4-s + 5-s + 1.43·6-s − 3.08·7-s + 8-s − 0.950·9-s + 10-s + 6.46·11-s + 1.43·12-s + 3.95·13-s − 3.08·14-s + 1.43·15-s + 16-s + 3.43·17-s − 0.950·18-s − 3.08·19-s + 20-s − 4.41·21-s + 6.46·22-s + 1.43·24-s + 25-s + 3.95·26-s − 5.65·27-s − 3.08·28-s + 0.863·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.826·3-s + 0.5·4-s + 0.447·5-s + 0.584·6-s − 1.16·7-s + 0.353·8-s − 0.316·9-s + 0.316·10-s + 1.95·11-s + 0.413·12-s + 1.09·13-s − 0.825·14-s + 0.369·15-s + 0.250·16-s + 0.832·17-s − 0.224·18-s − 0.708·19-s + 0.223·20-s − 0.964·21-s + 1.37·22-s + 0.292·24-s + 0.200·25-s + 0.774·26-s − 1.08·27-s − 0.583·28-s + 0.160·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.728350447\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.728350447\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 1.43T + 3T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 - 6.46T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 - 3.43T + 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 29 | \( 1 - 0.863T + 29T^{2} \) |
| 31 | \( 1 + 5.95T + 31T^{2} \) |
| 37 | \( 1 - 7.03T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 6.86T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−8.445940658847684707209292892214, −7.25622689977133795452585676619, −6.60186370564124759268677301951, −6.06684882040549659096926270195, −5.51032919257011054491803318402, −4.04559357599305318209509754599, −3.76286995378573907334876479545, −3.04415772202183749083416394996, −2.12394577228632665167922097117, −1.07533825837525047937364396011,
1.07533825837525047937364396011, 2.12394577228632665167922097117, 3.04415772202183749083416394996, 3.76286995378573907334876479545, 4.04559357599305318209509754599, 5.51032919257011054491803318402, 6.06684882040549659096926270195, 6.60186370564124759268677301951, 7.25622689977133795452585676619, 8.445940658847684707209292892214