| L(s) = 1 | + 2·2-s − 2.37·3-s + 4·4-s − 5·5-s − 4.74·6-s + 8·8-s − 21.3·9-s − 10·10-s + 49.8·11-s − 9.49·12-s − 42.1·13-s + 11.8·15-s + 16·16-s − 22.4·17-s − 42.7·18-s + 106.·19-s − 20·20-s + 99.7·22-s + 189.·23-s − 18.9·24-s + 25·25-s − 84.2·26-s + 114.·27-s − 52.5·29-s + 23.7·30-s − 154.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.456·3-s + 0.5·4-s − 0.447·5-s − 0.323·6-s + 0.353·8-s − 0.791·9-s − 0.316·10-s + 1.36·11-s − 0.228·12-s − 0.898·13-s + 0.204·15-s + 0.250·16-s − 0.320·17-s − 0.559·18-s + 1.28·19-s − 0.223·20-s + 0.966·22-s + 1.71·23-s − 0.161·24-s + 0.200·25-s − 0.635·26-s + 0.818·27-s − 0.336·29-s + 0.144·30-s − 0.893·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.375905077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.375905077\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.37T + 27T^{2} \) |
| 11 | \( 1 - 49.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 106.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 52.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 317.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 427.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 498.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 524.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 310.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 653.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 403.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 199.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 222.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 34.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 679.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 69.1T + 9.12e5T^{2} \) |
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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
−11.08763149252633696453521480541, −9.618917459621366206343498356270, −8.856772443487218172508433882136, −7.50955401793080132775368257487, −6.80851029057724163122488287598, −5.72670343306650941533389431512, −4.88811227988080340069432124386, −3.78264159215190633267759239332, −2.68139767443816221537740066963, −0.906990389138490854467633387508,
0.906990389138490854467633387508, 2.68139767443816221537740066963, 3.78264159215190633267759239332, 4.88811227988080340069432124386, 5.72670343306650941533389431512, 6.80851029057724163122488287598, 7.50955401793080132775368257487, 8.856772443487218172508433882136, 9.618917459621366206343498356270, 11.08763149252633696453521480541
