L(s) = 1 | + (−1 + 1.41i)3-s + (0.724 − 1.25i)5-s + (−0.5 − 0.866i)7-s + (−1.00 − 2.82i)9-s + (1 + 1.73i)11-s + (−2.44 + 4.24i)13-s + (1.05 + 2.28i)15-s + 2·17-s − 2.55·19-s + (1.72 + 0.158i)21-s + (−0.5 + 0.866i)23-s + (1.44 + 2.51i)25-s + (5.00 + 1.41i)27-s + (3.44 + 5.97i)29-s + (3 − 5.19i)31-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.816i)3-s + (0.324 − 0.561i)5-s + (−0.188 − 0.327i)7-s + (−0.333 − 0.942i)9-s + (0.301 + 0.522i)11-s + (−0.679 + 1.17i)13-s + (0.271 + 0.588i)15-s + 0.485·17-s − 0.585·19-s + (0.376 + 0.0346i)21-s + (−0.104 + 0.180i)23-s + (0.289 + 0.502i)25-s + (0.962 + 0.272i)27-s + (0.640 + 1.10i)29-s + (0.538 − 0.933i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00922 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00922 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.124138752\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.124138752\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.724 + 1.25i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.44 - 4.24i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 2.55T + 19T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.44 - 5.97i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + (4.89 - 8.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.44 - 5.97i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.89 - 8.48i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.27 + 5.67i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.44 - 11.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.101T + 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 + (0.949 + 1.64i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + (1.44 + 2.51i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.883685567065142100191570738768, −9.591954884081203779428225928537, −8.821767651070754941225395021555, −7.61861189371329348064761192178, −6.59346717162850579785892586816, −5.88901167290752387832275338558, −4.58615775155479881587904839052, −4.44962215031996023637117947626, −2.96158404149998557851368046115, −1.31698624389686944890217139504,
0.60164079668848799712035087560, 2.25013061593942333521244211182, 3.08851703960304538358922525791, 4.67130306694046274869808185284, 5.76642568124034498705433522459, 6.23042405133499405351376482239, 7.15958967631356796316169491940, 7.991579957621757819259544721453, 8.761235067579355374281873258742, 10.09396562082272593694789489336