L(s) = 1 | + 0.431i·3-s + i·5-s − 3.33·7-s + 2.81·9-s − 1.49·11-s + 5.60·13-s − 0.431·15-s − 0.587i·17-s + 0.133·19-s − 1.43i·21-s + (3.96 − 2.70i)23-s − 25-s + 2.51i·27-s − 1.54·29-s + 1.79i·31-s + ⋯ |
L(s) = 1 | + 0.249i·3-s + 0.447i·5-s − 1.25·7-s + 0.937·9-s − 0.450·11-s + 1.55·13-s − 0.111·15-s − 0.142i·17-s + 0.0306·19-s − 0.313i·21-s + (0.825 − 0.563i)23-s − 0.200·25-s + 0.483i·27-s − 0.286·29-s + 0.321i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.574031760\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.574031760\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-3.96 + 2.70i)T \) |
good | 3 | \( 1 - 0.431iT - 3T^{2} \) |
| 7 | \( 1 + 3.33T + 7T^{2} \) |
| 11 | \( 1 + 1.49T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 + 0.587iT - 17T^{2} \) |
| 19 | \( 1 - 0.133T + 19T^{2} \) |
| 29 | \( 1 + 1.54T + 29T^{2} \) |
| 31 | \( 1 - 1.79iT - 31T^{2} \) |
| 37 | \( 1 - 5.62iT - 37T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 - 5.67T + 43T^{2} \) |
| 47 | \( 1 - 6.73iT - 47T^{2} \) |
| 53 | \( 1 + 0.671iT - 53T^{2} \) |
| 59 | \( 1 - 3.46iT - 59T^{2} \) |
| 61 | \( 1 - 13.7iT - 61T^{2} \) |
| 67 | \( 1 - 3.10T + 67T^{2} \) |
| 71 | \( 1 - 12.2iT - 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 8.78T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 1.49iT - 89T^{2} \) |
| 97 | \( 1 + 1.91iT - 97T^{2} \) |
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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
−9.426096178169565780550326143445, −8.762692528146680185998374549314, −7.75779515135123366756947603115, −6.83092378365997501933882407274, −6.38942890036410876489046000067, −5.43654288674154602380223375961, −4.27051976753118522978871996427, −3.50236886525010431081341039022, −2.69222374942196819653989377588, −1.12926486325409796627501428871,
0.69923762420174064532279115556, 1.92770054758787869271947524396, 3.33730500243176400955722457631, 3.93231681120156240360406330043, 5.10928347143126275347566382130, 6.03887092371551360505576691688, 6.67125942982020017247917315802, 7.49943991147383719096290644298, 8.340204888643283058384925640642, 9.236604425223408798895746071857