| L(s) = 1 | − 0.801·2-s − 1.35·4-s − 0.246·5-s − 2.35·7-s + 2.69·8-s + 0.198·10-s + 4.24·11-s + 1.89·14-s + 0.554·16-s − 2.15·17-s − 0.0881·19-s + 0.335·20-s − 3.40·22-s − 1.49·23-s − 4.93·25-s + 3.19·28-s − 4.63·29-s − 6.63·31-s − 5.82·32-s + 1.73·34-s + 0.582·35-s + 5.69·37-s + 0.0706·38-s − 0.664·40-s + 11.5·41-s − 0.295·43-s − 5.76·44-s + ⋯ |
| L(s) = 1 | − 0.567·2-s − 0.678·4-s − 0.110·5-s − 0.890·7-s + 0.951·8-s + 0.0626·10-s + 1.28·11-s + 0.505·14-s + 0.138·16-s − 0.523·17-s − 0.0202·19-s + 0.0749·20-s − 0.726·22-s − 0.311·23-s − 0.987·25-s + 0.604·28-s − 0.859·29-s − 1.19·31-s − 1.03·32-s + 0.296·34-s + 0.0983·35-s + 0.935·37-s + 0.0114·38-s − 0.105·40-s + 1.81·41-s − 0.0451·43-s − 0.868·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8232036574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8232036574\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.801T + 2T^{2} \) |
| 5 | \( 1 + 0.246T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 + 0.0881T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 + 4.63T + 29T^{2} \) |
| 31 | \( 1 + 6.63T + 31T^{2} \) |
| 37 | \( 1 - 5.69T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 0.295T + 43T^{2} \) |
| 47 | \( 1 - 7.35T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 61 | \( 1 - 3.47T + 61T^{2} \) |
| 67 | \( 1 - 7.67T + 67T^{2} \) |
| 71 | \( 1 - 8.66T + 71T^{2} \) |
| 73 | \( 1 - 6.73T + 73T^{2} \) |
| 79 | \( 1 - 9.97T + 79T^{2} \) |
| 83 | \( 1 + 1.60T + 83T^{2} \) |
| 89 | \( 1 - 2.88T + 89T^{2} \) |
| 97 | \( 1 + 8.05T + 97T^{2} \) |
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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
−9.347983437019907820603351928706, −8.970872501699235797086285806354, −7.941835166850545149808675327616, −7.17814612996643894868353930346, −6.28306691977423484462581585657, −5.42756834852429223085532035726, −4.07407053636868891309488860625, −3.78013333863930320822010066069, −2.13962796236265663010365082947, −0.70386815798282751834484456238,
0.70386815798282751834484456238, 2.13962796236265663010365082947, 3.78013333863930320822010066069, 4.07407053636868891309488860625, 5.42756834852429223085532035726, 6.28306691977423484462581585657, 7.17814612996643894868353930346, 7.941835166850545149808675327616, 8.970872501699235797086285806354, 9.347983437019907820603351928706
