| L(s) = 1 | + (1 − i)3-s + (−1 − i)7-s + i·9-s + 6i·11-s + (−1 − i)13-s + (−1 + i)17-s + 4·19-s − 2·21-s + (5 − 5i)23-s + (4 + 4i)27-s + 8i·29-s + 2i·31-s + (6 + 6i)33-s + (−5 + 5i)37-s − 2·39-s + ⋯ |
| L(s) = 1 | + (0.577 − 0.577i)3-s + (−0.377 − 0.377i)7-s + 0.333i·9-s + 1.80i·11-s + (−0.277 − 0.277i)13-s + (−0.242 + 0.242i)17-s + 0.917·19-s − 0.436·21-s + (1.04 − 1.04i)23-s + (0.769 + 0.769i)27-s + 1.48i·29-s + 0.359i·31-s + (1.04 + 1.04i)33-s + (−0.821 + 0.821i)37-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.881964884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.881964884\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-1 + i)T - 3iT^{2} \) |
| 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 - 6iT - 11T^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + (1 - i)T - 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-5 + 5i)T - 23iT^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7 - 7i)T + 47iT^{2} \) |
| 53 | \( 1 + (1 + i)T + 53iT^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (-7 - 7i)T + 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (9 + 9i)T + 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (-5 + 5i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (-3 + 3i)T - 97iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.399138532315601929631861252584, −8.683378625186750792913895541591, −7.69670304685218610623931435606, −7.16307409618371448895878095344, −6.62869100103402877147323772212, −5.15604000074851863198251272913, −4.59793702833499293242457194665, −3.28253317423061125992855079629, −2.39312566231043210505475636148, −1.33689939001090117967721838301,
0.73496640570089924081057845686, 2.55911487843505104698930491476, 3.34403631815961901670637454084, 4.01837887188466131351765351014, 5.33786599831243196001657380320, 5.95393472857215121955830003603, 6.93727411485892768383615336976, 7.908526189653864853575777906300, 8.797798566465001675462334830021, 9.243072371392499420253651797293
