| L(s) = 1 | + i·3-s − 26i·7-s + 26·9-s − 45·11-s − 44i·13-s + 117i·17-s − 91·19-s + 26·21-s − 18i·23-s + 53i·27-s − 144·29-s − 26·31-s − 45i·33-s − 214i·37-s + 44·39-s + ⋯ |
| L(s) = 1 | + 0.192i·3-s − 1.40i·7-s + 0.962·9-s − 1.23·11-s − 0.938i·13-s + 1.66i·17-s − 1.09·19-s + 0.270·21-s − 0.163i·23-s + 0.377i·27-s − 0.922·29-s − 0.150·31-s − 0.237i·33-s − 0.950i·37-s + 0.180·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6709085638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6709085638\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - iT - 27T^{2} \) |
| 7 | \( 1 + 26iT - 343T^{2} \) |
| 11 | \( 1 + 45T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 117iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 91T + 6.85e3T^{2} \) |
| 23 | \( 1 + 18iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 144T + 2.43e4T^{2} \) |
| 31 | \( 1 + 26T + 2.97e4T^{2} \) |
| 37 | \( 1 + 214iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 459T + 6.89e4T^{2} \) |
| 43 | \( 1 + 460iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 468iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 558iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 72T + 2.05e5T^{2} \) |
| 61 | \( 1 + 118T + 2.26e5T^{2} \) |
| 67 | \( 1 + 251iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 108T + 3.57e5T^{2} \) |
| 73 | \( 1 + 299iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 898T + 4.93e5T^{2} \) |
| 83 | \( 1 - 927iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 351T + 7.04e5T^{2} \) |
| 97 | \( 1 - 386iT - 9.12e5T^{2} \) |
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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
−10.56274284429775148549756514350, −9.922413046763944970091042655067, −8.427460416979262564376503711780, −7.67919442404474539100835531862, −6.82256444712194132480159139523, −5.55708454945625139759102586587, −4.35926437920469053597212026933, −3.55242601428833158527110280667, −1.79458506374998210883146754356, −0.21101499771311045873645691305,
1.85758253460488688080849070393, 2.85166632103282400950275483469, 4.54329257044329439848044059741, 5.39723636120386282890234038006, 6.57221334238163972215349957233, 7.48651968093925570464671610454, 8.555030750358521281633070061985, 9.389966774357262107591964086596, 10.22952993191318345558407880096, 11.42719545365814315039984594559
