L(s) = 1 | + (−1 + 1.73i)5-s + (1 + 1.73i)7-s + (1 + 1.73i)11-s + (0.5 − 0.866i)13-s + 6·17-s − 6·19-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s + (−1 − 1.73i)29-s + (−5 + 8.66i)31-s − 3.99·35-s − 6·37-s + (3 − 5.19i)41-s + (−2 − 3.46i)43-s + (1 + 1.73i)47-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.774i)5-s + (0.377 + 0.654i)7-s + (0.301 + 0.522i)11-s + (0.138 − 0.240i)13-s + 1.45·17-s − 1.37·19-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s + (−0.185 − 0.321i)29-s + (−0.898 + 1.55i)31-s − 0.676·35-s − 0.986·37-s + (0.468 − 0.811i)41-s + (−0.304 − 0.528i)43-s + (0.145 + 0.252i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.127310944\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.127310944\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.624265590424849915600391747664, −7.980767723666915448596498309910, −7.23597426317553315411605002760, −6.70444702205413018812873600587, −5.59472594954575488756539201543, −5.26480494130717158267404582058, −3.90550110446797080852529747824, −3.50573402356748327702214774980, −2.37240852920154976805993248198, −1.50184267658733535969875812488,
0.33041763291423011087851613387, 1.29819833165106485558294966642, 2.44293332472713963989103048550, 3.76899013665345692418308127054, 4.16349127212245177178904084549, 4.97671880569526842381040977291, 5.90201570534229738204400098804, 6.55802763671425118560280122004, 7.55089957796345854712786447494, 8.128286327613359879013789263651