| L(s) = 1 | − 3.26·3-s + 5-s + 4.40i·7-s + 7.69·9-s + 4.40i·11-s + 4.96i·13-s − 3.26·15-s + (3.26 + 2.88i)19-s − 14.3i·21-s − 7.44i·23-s + 25-s − 15.3·27-s + 8.79·31-s − 14.3i·33-s + 4.40i·35-s + ⋯ |
| L(s) = 1 | − 1.88·3-s + 0.447·5-s + 1.66i·7-s + 2.56·9-s + 1.32i·11-s + 1.37i·13-s − 0.844·15-s + (0.750 + 0.661i)19-s − 3.14i·21-s − 1.55i·23-s + 0.200·25-s − 2.95·27-s + 1.57·31-s − 2.50i·33-s + 0.744i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8675246008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8675246008\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + (-3.26 - 2.88i)T \) |
| good | 3 | \( 1 + 3.26T + 3T^{2} \) |
| 7 | \( 1 - 4.40iT - 7T^{2} \) |
| 11 | \( 1 - 4.40iT - 11T^{2} \) |
| 13 | \( 1 - 4.96iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 7.44iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 8.79T + 31T^{2} \) |
| 37 | \( 1 - 9.42iT - 37T^{2} \) |
| 41 | \( 1 + 4.45iT - 41T^{2} \) |
| 43 | \( 1 - 1.36iT - 43T^{2} \) |
| 47 | \( 1 - 4.40iT - 47T^{2} \) |
| 53 | \( 1 + 0.512iT - 53T^{2} \) |
| 59 | \( 1 - 2.25T + 59T^{2} \) |
| 61 | \( 1 - 4.69T + 61T^{2} \) |
| 67 | \( 1 + 1.01T + 67T^{2} \) |
| 71 | \( 1 + 2.25T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 8.79T + 79T^{2} \) |
| 83 | \( 1 + 10.4iT - 83T^{2} \) |
| 89 | \( 1 - 8.91iT - 89T^{2} \) |
| 97 | \( 1 + 0.512iT - 97T^{2} \) |
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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
−9.841332047661720666678592251327, −9.296821898848611669316983045834, −8.169712406176187627235320545365, −6.83707970997238168398743899000, −6.48752984080856161698725641080, −5.70134555275968570368057657875, −4.90993322302946339793870455465, −4.38751115318705136735496936768, −2.43204726924672524398763234086, −1.46803868187257182966546521988,
0.55497552670202221319241987952, 1.11827037669532715648079265461, 3.28371731527008943708717974531, 4.24518443307198200314755380592, 5.32624949602905324124668426120, 5.69063249424357353069715878951, 6.64735209527542732338517987422, 7.30536788475463473555326082793, 8.093080825092200665445580583080, 9.620178570415436406155989515029
