L(s) = 1 | − 2-s + (−1.5 − 0.866i)3-s + 4-s + (3 + 1.73i)5-s + (1.5 + 0.866i)6-s + (−0.5 − 2.59i)7-s − 8-s + (1.5 + 2.59i)9-s + (−3 − 1.73i)10-s + (3 − 5.19i)11-s + (−1.5 − 0.866i)12-s + (−2.5 + 2.59i)13-s + (0.5 + 2.59i)14-s + (−3 − 5.19i)15-s + 16-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.866 − 0.499i)3-s + 0.5·4-s + (1.34 + 0.774i)5-s + (0.612 + 0.353i)6-s + (−0.188 − 0.981i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.948 − 0.547i)10-s + (0.904 − 1.56i)11-s + (−0.433 − 0.249i)12-s + (−0.693 + 0.720i)13-s + (0.133 + 0.694i)14-s + (−0.774 − 1.34i)15-s + 0.250·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.945841 - 0.353649i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.945841 - 0.353649i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
| 13 | \( 1 + (2.5 - 2.59i)T \) |
good | 5 | \( 1 + (-3 - 1.73i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + (-6 + 3.46i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8.66iT - 37T^{2} \) |
| 41 | \( 1 + (-9 + 5.19i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6 - 3.46i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 1.73i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 3.46iT - 89T^{2} \) |
| 97 | \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.73378141141859610697875123829, −9.875339453351827224945861881078, −9.255990877082562062631749253532, −7.77214652439216223516170469366, −7.00967729811912764498299557089, −6.16723364235994099169436458262, −5.69357377689222093803945954762, −3.85227911136320157557543189351, −2.25939069747099897410742993062, −0.977735717229104235874247603134,
1.29881398216898491119744554424, 2.63357097597375079113730642612, 4.72821050387656558243373913926, 5.29053477758604399199797271050, 6.33561102252767572507756785780, 7.06738395159695094126951902046, 8.712119187393681288506658530560, 9.362358875780042720200944838944, 9.843691173542517022588394941901, 10.56230978220468315845599432778