| L(s) = 1 | + 9i·3-s − 26i·7-s − 54·9-s − 59·11-s + 28i·13-s − 5i·17-s − 109·19-s + 234·21-s − 194i·23-s − 243i·27-s + 32·29-s + 10·31-s − 531i·33-s + 198i·37-s − 252·39-s + ⋯ |
| L(s) = 1 | + 1.73i·3-s − 1.40i·7-s − 2·9-s − 1.61·11-s + 0.597i·13-s − 0.0713i·17-s − 1.31·19-s + 2.43·21-s − 1.75i·23-s − 1.73i·27-s + 0.204·29-s + 0.0579·31-s − 2.80i·33-s + 0.879i·37-s − 1.03·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 9iT - 27T^{2} \) |
| 7 | \( 1 + 26iT - 343T^{2} \) |
| 11 | \( 1 + 59T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 5iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 109T + 6.85e3T^{2} \) |
| 23 | \( 1 + 194iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 32T + 2.43e4T^{2} \) |
| 31 | \( 1 - 10T + 2.97e4T^{2} \) |
| 37 | \( 1 - 198iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 117T + 6.89e4T^{2} \) |
| 43 | \( 1 - 388iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 68iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 18iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 392T + 2.05e5T^{2} \) |
| 61 | \( 1 + 710T + 2.26e5T^{2} \) |
| 67 | \( 1 - 253iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 612T + 3.57e5T^{2} \) |
| 73 | \( 1 + 549iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 414T + 4.93e5T^{2} \) |
| 83 | \( 1 + 121iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 81T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.50e3iT - 9.12e5T^{2} \) |
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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
−11.15393938352522419346794942557, −10.49402007663718025376895868752, −10.11515541157881633164754281975, −8.845914893732051975166137237585, −7.82334988108034490830774446384, −6.31954877183554949997475755517, −4.74456189187046857897397817155, −4.30213702937028984728472696706, −2.86651536518036680132163365638, 0,
1.93969814527912622473200520647, 2.83505152557841791525048598942, 5.40348380460050031519800036264, 6.01862108603600452737511220259, 7.38160533140963653868149082931, 8.104698253017225594076831966140, 8.979985282582842188120706929513, 10.57688960264807055982568372232, 11.67348721665217986390393719824
