L(s) = 1 | + (0.200 + 0.979i)2-s + (−0.919 + 0.391i)4-s + (−0.996 − 0.0804i)5-s + (−0.568 − 0.822i)8-s + (−0.948 − 0.316i)9-s + (−0.120 − 0.992i)10-s + (0.428 + 0.903i)13-s + (0.692 − 0.721i)16-s + (−0.708 − 0.336i)17-s + (0.120 − 0.992i)18-s + (0.948 − 0.316i)20-s + (0.987 + 0.160i)25-s + (−0.799 + 0.600i)26-s + (−0.253 − 1.23i)29-s + (0.845 + 0.534i)32-s + ⋯ |
L(s) = 1 | + (0.200 + 0.979i)2-s + (−0.919 + 0.391i)4-s + (−0.996 − 0.0804i)5-s + (−0.568 − 0.822i)8-s + (−0.948 − 0.316i)9-s + (−0.120 − 0.992i)10-s + (0.428 + 0.903i)13-s + (0.692 − 0.721i)16-s + (−0.708 − 0.336i)17-s + (0.120 − 0.992i)18-s + (0.948 − 0.316i)20-s + (0.987 + 0.160i)25-s + (−0.799 + 0.600i)26-s + (−0.253 − 1.23i)29-s + (0.845 + 0.534i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6652724996\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6652724996\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.200 - 0.979i)T \) |
| 5 | \( 1 + (0.996 + 0.0804i)T \) |
| 13 | \( 1 + (-0.428 - 0.903i)T \) |
good | 3 | \( 1 + (0.948 + 0.316i)T^{2} \) |
| 7 | \( 1 + (0.632 + 0.774i)T^{2} \) |
| 11 | \( 1 + (0.799 - 0.600i)T^{2} \) |
| 17 | \( 1 + (0.708 + 0.336i)T + (0.632 + 0.774i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.253 + 1.23i)T + (-0.919 + 0.391i)T^{2} \) |
| 31 | \( 1 + (0.568 + 0.822i)T^{2} \) |
| 37 | \( 1 + (-0.470 + 0.297i)T + (0.428 - 0.903i)T^{2} \) |
| 41 | \( 1 + (-0.314 + 1.93i)T + (-0.948 - 0.316i)T^{2} \) |
| 43 | \( 1 + (0.428 + 0.903i)T^{2} \) |
| 47 | \( 1 + (0.970 - 0.239i)T^{2} \) |
| 53 | \( 1 + (-1.48 + 1.02i)T + (0.354 - 0.935i)T^{2} \) |
| 59 | \( 1 + (-0.0402 + 0.999i)T^{2} \) |
| 61 | \( 1 + (-1.96 - 0.158i)T + (0.987 + 0.160i)T^{2} \) |
| 67 | \( 1 + (-0.692 - 0.721i)T^{2} \) |
| 71 | \( 1 + (-0.200 - 0.979i)T^{2} \) |
| 73 | \( 1 + (0.748 + 0.663i)T + (0.120 + 0.992i)T^{2} \) |
| 79 | \( 1 + (0.970 - 0.239i)T^{2} \) |
| 83 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 89 | \( 1 + (0.414 + 0.239i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.492 - 1.70i)T + (-0.845 + 0.534i)T^{2} \) |
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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
−8.696322874206270573427798459990, −8.043337399957266289385007363132, −7.19672470733182741091235117580, −6.65731479934721743637122333590, −5.82437225799122899578780193097, −5.04852115598010845637782084529, −4.07589690447188411102249082511, −3.69043930292156999138199451770, −2.45091712742915131627907118385, −0.43888802582719246932221577034,
1.10120311175647800795810699833, 2.56131852359498838448736022643, 3.17928803946303708917234022351, 3.99949720607879409312834095740, 4.82244851592521652255139897935, 5.58183279927911802603180002320, 6.42863567995705433959675737371, 7.56168772789205643126009623233, 8.357529953597616178110458913874, 8.651668995955153975298292835373