L(s) = 1 | + (1.26 − 2.19i)5-s + (−1.98 − 1.74i)7-s + (−5.68 + 3.28i)11-s + (−5.13 − 2.96i)13-s + 3.52·17-s + 0.261i·19-s + (2.89 + 1.67i)23-s + (−0.718 − 1.24i)25-s + (1.00 − 0.582i)29-s + (1.69 + 0.977i)31-s + (−6.36 + 2.14i)35-s − 1.16·37-s + (−2.85 + 4.94i)41-s + (−2.67 − 4.63i)43-s + (3.79 + 6.57i)47-s + ⋯ |
L(s) = 1 | + (0.567 − 0.982i)5-s + (−0.750 − 0.660i)7-s + (−1.71 + 0.989i)11-s + (−1.42 − 0.822i)13-s + 0.855·17-s + 0.0599i·19-s + (0.604 + 0.349i)23-s + (−0.143 − 0.248i)25-s + (0.187 − 0.108i)29-s + (0.304 + 0.175i)31-s + (−1.07 + 0.362i)35-s − 0.191·37-s + (−0.446 + 0.773i)41-s + (−0.407 − 0.706i)43-s + (0.553 + 0.959i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0303 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0303 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5256182972\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5256182972\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.98 + 1.74i)T \) |
good | 5 | \( 1 + (-1.26 + 2.19i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (5.68 - 3.28i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.13 + 2.96i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 - 0.261iT - 19T^{2} \) |
| 23 | \( 1 + (-2.89 - 1.67i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.00 + 0.582i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.69 - 0.977i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.16T + 37T^{2} \) |
| 41 | \( 1 + (2.85 - 4.94i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.67 + 4.63i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.79 - 6.57i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 3.54iT - 53T^{2} \) |
| 59 | \( 1 + (5.47 - 9.48i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.53 + 3.19i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.54 - 7.86i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.5iT - 71T^{2} \) |
| 73 | \( 1 - 2.72iT - 73T^{2} \) |
| 79 | \( 1 + (-0.652 - 1.13i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.53 - 7.85i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + (9.95 - 5.74i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.106089632054895974291495750066, −7.87475322190844402576662640222, −7.63289812820673555753348531404, −6.76649395786956916269507281632, −5.55838325280929864770269457948, −5.15296713430213295879087301968, −4.47988428995947024712582013719, −3.14183452754613344521148075161, −2.42913506920190213541089787682, −1.08773397207064079276764397034,
0.17130933159710709336321805061, 2.20915163025225859849140538458, 2.75814051583335132412673878165, 3.40351822420601612691143465796, 4.92329304102905426545270682697, 5.47776472685774807289264901401, 6.29814231031222729670055495831, 6.94718560065827491507758619449, 7.70927341074002119965736509581, 8.548875216677354236327242032021