| L(s) = 1 | + 2·2-s + 8·3-s + 4·4-s + 5·5-s + 16·6-s − 12·7-s + 8·8-s + 37·9-s + 10·10-s − 11·11-s + 32·12-s − 34·13-s − 24·14-s + 40·15-s + 16·16-s − 86·17-s + 74·18-s − 4·19-s + 20·20-s − 96·21-s − 22·22-s + 148·23-s + 64·24-s + 25·25-s − 68·26-s + 80·27-s − 48·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s + 1/2·4-s + 0.447·5-s + 1.08·6-s − 0.647·7-s + 0.353·8-s + 1.37·9-s + 0.316·10-s − 0.301·11-s + 0.769·12-s − 0.725·13-s − 0.458·14-s + 0.688·15-s + 1/4·16-s − 1.22·17-s + 0.968·18-s − 0.0482·19-s + 0.223·20-s − 0.997·21-s − 0.213·22-s + 1.34·23-s + 0.544·24-s + 1/5·25-s − 0.512·26-s + 0.570·27-s − 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.594102952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.594102952\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 86 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 148 T + p^{3} T^{2} \) |
| 29 | \( 1 - 134 T + p^{3} T^{2} \) |
| 31 | \( 1 + 280 T + p^{3} T^{2} \) |
| 37 | \( 1 - 430 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 136 T + p^{3} T^{2} \) |
| 47 | \( 1 + 28 T + p^{3} T^{2} \) |
| 53 | \( 1 + 658 T + p^{3} T^{2} \) |
| 59 | \( 1 - 4 T + p^{3} T^{2} \) |
| 61 | \( 1 + 90 T + p^{3} T^{2} \) |
| 67 | \( 1 - 96 T + p^{3} T^{2} \) |
| 71 | \( 1 - 816 T + p^{3} T^{2} \) |
| 73 | \( 1 + 430 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1296 T + p^{3} T^{2} \) |
| 83 | \( 1 + 608 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 706 T + p^{3} T^{2} \) |
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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
−13.15106218773735419196510859184, −12.82478064022648878307882670295, −11.08307414186094850880382218359, −9.750867972880572090575287365812, −8.952358325740457934369857929606, −7.62770120649784248680484115488, −6.50431227374710379533789534864, −4.76139229998996872602964402370, −3.25610859441392328566027158661, −2.25984003118220183730307166482,
2.25984003118220183730307166482, 3.25610859441392328566027158661, 4.76139229998996872602964402370, 6.50431227374710379533789534864, 7.62770120649784248680484115488, 8.952358325740457934369857929606, 9.750867972880572090575287365812, 11.08307414186094850880382218359, 12.82478064022648878307882670295, 13.15106218773735419196510859184
