L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.654 − 0.755i)3-s + (0.415 − 0.909i)4-s + (−0.142 + 0.989i)6-s − 0.563i·7-s + (0.142 + 0.989i)8-s + (−0.142 − 0.989i)9-s + 0.830·11-s + (−0.415 − 0.909i)12-s + (0.304 + 0.474i)14-s + (−0.654 − 0.755i)16-s + (0.654 + 0.755i)18-s − 1.97i·19-s + (−0.425 − 0.368i)21-s + (−0.698 + 0.449i)22-s + ⋯ |
L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.654 − 0.755i)3-s + (0.415 − 0.909i)4-s + (−0.142 + 0.989i)6-s − 0.563i·7-s + (0.142 + 0.989i)8-s + (−0.142 − 0.989i)9-s + 0.830·11-s + (−0.415 − 0.909i)12-s + (0.304 + 0.474i)14-s + (−0.654 − 0.755i)16-s + (0.654 + 0.755i)18-s − 1.97i·19-s + (−0.425 − 0.368i)21-s + (−0.698 + 0.449i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9671678534\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9671678534\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + 0.563iT - T^{2} \) |
| 11 | \( 1 - 0.830T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.97iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 0.563iT - T^{2} \) |
| 31 | \( 1 - 1.08iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + 0.284T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.30T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.51iT - T^{2} \) |
| 97 | \( 1 - 0.830T + T^{2} \) |
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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
−9.036483486370348433608526900251, −8.485688080511114843893043592398, −7.58402296346671488965268919003, −6.93176771893214279065774499160, −6.55490683984774063357918951251, −5.44112824327563379821551036371, −4.30529357678363594467503717357, −3.09165506851289350780572416938, −1.97698034679562403826132096125, −0.888652500604570966108894786304,
1.69381371063382131859967947326, 2.53710441603647069526241750672, 3.72955240036762938118973786302, 4.07559925075590865413829648429, 5.52661232692583232752724931819, 6.38497130008970915528005010919, 7.62630823156808089667672893108, 8.119646495568487891803084921884, 8.829751826193278339525473268198, 9.605630151678899557260132132961