| L(s) = 1 | − 8i·3-s + 108i·7-s + 179·9-s + 604·11-s + 306i·13-s + 930i·17-s − 1.32e3·19-s + 864·21-s − 852i·23-s − 3.37e3i·27-s − 5.90e3·29-s + 3.32e3·31-s − 4.83e3i·33-s + 1.07e4i·37-s + 2.44e3·39-s + ⋯ |
| L(s) = 1 | − 0.513i·3-s + 0.833i·7-s + 0.736·9-s + 1.50·11-s + 0.502i·13-s + 0.780i·17-s − 0.841·19-s + 0.427·21-s − 0.335i·23-s − 0.891i·27-s − 1.30·29-s + 0.620·31-s − 0.772i·33-s + 1.29i·37-s + 0.257·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.091219188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091219188\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 8iT - 243T^{2} \) |
| 7 | \( 1 - 108iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 604T + 1.61e5T^{2} \) |
| 13 | \( 1 - 306iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 930iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.32e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 852iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.90e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.07e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.79e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.26e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 9.79e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.14e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.88e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.72e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.45e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.54e5iT - 8.58e9T^{2} \) |
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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
−10.65445035125637801813720762235, −9.546362931279703708393155210001, −8.835494624089443193764362778523, −7.86668449649405892437048487962, −6.61069484597969621770665758270, −6.23505761028207078459364958855, −4.68960456417940979107240878305, −3.70012032248649133552305494802, −2.10416239428805585811590575855, −1.26660297723123933781970023821,
0.54332976580278596911369038418, 1.78791649668315935550534200206, 3.60387233109569261711995451988, 4.16597265225463421745655187640, 5.33828625764272842774081439359, 6.71804863205368340943235155279, 7.30302542054420591552798372388, 8.613023420498874296604025759046, 9.534214231269797203751912698221, 10.21607833990852704940050712860
