L(s) = 1 | + 2.41i·3-s + (2.12 + 0.707i)5-s + 2.41i·7-s − 2.82·9-s − 1.41·11-s − 1.82i·13-s + (−1.70 + 5.12i)15-s + i·17-s − 19-s − 5.82·21-s + 5.24i·23-s + (3.99 + 3i)25-s + 0.414i·27-s − 3.82·29-s − 3.41·31-s + ⋯ |
L(s) = 1 | + 1.39i·3-s + (0.948 + 0.316i)5-s + 0.912i·7-s − 0.942·9-s − 0.426·11-s − 0.507i·13-s + (−0.440 + 1.32i)15-s + 0.242i·17-s − 0.229·19-s − 1.27·21-s + 1.09i·23-s + (0.799 + 0.600i)25-s + 0.0797i·27-s − 0.710·29-s − 0.613·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.670319909\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.670319909\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.12 - 0.707i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2.41iT - 3T^{2} \) |
| 7 | \( 1 - 2.41iT - 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 1.82iT - 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 23 | \( 1 - 5.24iT - 23T^{2} \) |
| 29 | \( 1 + 3.82T + 29T^{2} \) |
| 31 | \( 1 + 3.41T + 31T^{2} \) |
| 37 | \( 1 - 5.17iT - 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 2.24iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 1.82iT - 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 3.41T + 61T^{2} \) |
| 67 | \( 1 - 6.07iT - 67T^{2} \) |
| 71 | \( 1 + 3.07T + 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 0.828T + 79T^{2} \) |
| 83 | \( 1 + 2.48iT - 83T^{2} \) |
| 89 | \( 1 - 3.75T + 89T^{2} \) |
| 97 | \( 1 - 7.65iT - 97T^{2} \) |
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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
−10.00625112711815264058368887428, −9.115092753433907488094562286481, −8.627493739726744200707091368070, −7.45474023547109487275819863749, −6.31443354785152808386557464951, −5.38197193791193878841825432983, −5.17451060186451044927248202297, −3.80605261575253879895754112899, −2.98928491061079411314799678318, −1.91332638754287278901993752509,
0.63338995933067839325469605499, 1.74327967522851569861427311134, 2.52657008241130813305408273596, 4.00902587571035455570869971860, 5.11127440523543769365541882968, 6.02828787343619651952209657033, 6.81818591236742531507530799066, 7.30023868640796392684681351291, 8.226114675530389908845938839060, 8.977997089393732275069436107736