| L(s) = 1 | + (1 + i)2-s + 3-s + 2i·4-s + (1 + i)6-s + 2i·7-s + (−2 + 2i)8-s + 9-s + 4i·11-s + 2i·12-s + (−2 + 2i)14-s − 4·16-s − 6i·17-s + (1 + i)18-s − 4i·19-s + 2i·21-s + (−4 + 4i)22-s + ⋯ |
| L(s) = 1 | + (0.707 + 0.707i)2-s + 0.577·3-s + i·4-s + (0.408 + 0.408i)6-s + 0.755i·7-s + (−0.707 + 0.707i)8-s + 0.333·9-s + 1.20i·11-s + 0.577i·12-s + (−0.534 + 0.534i)14-s − 16-s − 1.45i·17-s + (0.235 + 0.235i)18-s − 0.917i·19-s + 0.436i·21-s + (−0.852 + 0.852i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43508 + 1.99107i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43508 + 1.99107i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−11.17280213434441775729354164527, −9.662488699462513012352011175620, −9.138990688243248336757955051629, −8.129832662337429748606819928419, −7.25484778424901926075178841069, −6.58055249715388620260134079843, −5.20591690215189884082556029488, −4.63689501371715664178488111216, −3.22959532377526981731033881577, −2.30106681279212984081645907654,
1.12033754783572824717961723555, 2.60573082288179315085877908054, 3.72465792120125972765801956572, 4.36377315028797765063199726698, 5.83440828607214858299831450367, 6.50607490609173362501205439226, 7.937331385299339575120187278410, 8.640995296333488094638450261966, 9.862013650422922002899832201568, 10.46558207188909209278887373858
