L(s) = 1 | − 2-s − 3-s + 6-s − 7-s + 8-s + 14-s − 16-s − 19-s + 21-s − 23-s − 24-s + 25-s + 27-s + 2·31-s + 2·37-s + 38-s + 2·41-s − 42-s − 43-s + 46-s − 47-s + 48-s − 50-s − 53-s − 54-s − 56-s + 57-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 6-s − 7-s + 8-s + 14-s − 16-s − 19-s + 21-s − 23-s − 24-s + 25-s + 27-s + 2·31-s + 2·37-s + 38-s + 2·41-s − 42-s − 43-s + 46-s − 47-s + 48-s − 50-s − 53-s − 54-s − 56-s + 57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2962805454\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2962805454\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 983 | \( 1 - T \) |
good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−10.00712953733909617276992179666, −9.616755537539473282462535986743, −8.532612663755738900708185302838, −7.932870312774793365563649268329, −6.59742462789750009560290066902, −6.27281409598544499559901967757, −5.02100310790343134337479493046, −4.11887443096406414322776565658, −2.59040587304164220179951415732, −0.75690031708046595279438787245,
0.75690031708046595279438787245, 2.59040587304164220179951415732, 4.11887443096406414322776565658, 5.02100310790343134337479493046, 6.27281409598544499559901967757, 6.59742462789750009560290066902, 7.932870312774793365563649268329, 8.532612663755738900708185302838, 9.616755537539473282462535986743, 10.00712953733909617276992179666