| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 3·7-s − 8-s + 9-s − 10-s + 4·11-s + 12-s + 13-s + 3·14-s + 15-s + 16-s − 3·17-s − 18-s − 4·19-s + 20-s − 3·21-s − 4·22-s − 3·23-s − 24-s + 25-s − 26-s + 27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.654·21-s − 0.852·22-s − 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.030097678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030097678\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.61884171166484, −12.31237129133725, −11.42385363156051, −11.32713417426404, −10.68169694445640, −10.13439168366944, −9.775724241919132, −9.381376072216617, −8.992779435372097, −8.630101731377285, −8.173042841464901, −7.505418015342097, −6.981278241180586, −6.615960172251302, −6.181255404154209, −5.903681661663458, −5.037472451216470, −4.383640792577342, −3.830497871847662, −3.416697502901539, −2.918528277522991, −2.057235179487692, −1.908940175291774, −1.163634048011649, −0.2870095142621159,
0.2870095142621159, 1.163634048011649, 1.908940175291774, 2.057235179487692, 2.918528277522991, 3.416697502901539, 3.830497871847662, 4.383640792577342, 5.037472451216470, 5.903681661663458, 6.181255404154209, 6.615960172251302, 6.981278241180586, 7.505418015342097, 8.173042841464901, 8.630101731377285, 8.992779435372097, 9.381376072216617, 9.775724241919132, 10.13439168366944, 10.68169694445640, 11.32713417426404, 11.42385363156051, 12.31237129133725, 12.61884171166484
