| L(s) = 1 | − 4.37i·3-s + 3.17·7-s − 10.1·9-s − 21.9i·11-s + 10.7i·13-s + 29.8i·17-s + 26.6·19-s − 13.8i·21-s + 9.73i·23-s + 5.16i·27-s − 12.5i·29-s + (−27.6 − 14.1i)31-s − 95.9·33-s − 48.4i·37-s + 46.8·39-s + ⋯ |
| L(s) = 1 | − 1.45i·3-s + 0.453·7-s − 1.13·9-s − 1.99i·11-s + 0.823i·13-s + 1.75i·17-s + 1.40·19-s − 0.661i·21-s + 0.423i·23-s + 0.191i·27-s − 0.434i·29-s + (−0.890 − 0.454i)31-s − 2.90·33-s − 1.31i·37-s + 1.20·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.232540499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.232540499\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + (27.6 + 14.1i)T \) |
| good | 3 | \( 1 + 4.37iT - 9T^{2} \) |
| 7 | \( 1 - 3.17T + 49T^{2} \) |
| 11 | \( 1 + 21.9iT - 121T^{2} \) |
| 13 | \( 1 - 10.7iT - 169T^{2} \) |
| 17 | \( 1 - 29.8iT - 289T^{2} \) |
| 19 | \( 1 - 26.6T + 361T^{2} \) |
| 23 | \( 1 - 9.73iT - 529T^{2} \) |
| 29 | \( 1 + 12.5iT - 841T^{2} \) |
| 37 | \( 1 + 48.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 68.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 36.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 5.79T + 2.20e3T^{2} \) |
| 53 | \( 1 - 11.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 18.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 97.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 88.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 28.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 27.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 78.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 37.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 123. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 102.T + 9.40e3T^{2} \) |
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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
−8.087676311810627331129480111275, −7.39815536761396935439644998507, −6.59435678838927087769094587877, −5.90382334419141582253180694124, −5.39824807854034093727283716455, −3.93584623533896461602684812568, −3.22579779678031780359773233393, −1.96072484985794731423966470610, −1.34852667670741822220141507619, −0.27294155976153291854298917857,
1.39946745507790491589792070363, 2.73871270225158466968824656901, 3.43459456125769737008101258721, 4.61770695870401944021163054303, 4.89260945862531959256397762704, 5.46024066242421868682647750324, 6.92255104049891166219598743715, 7.38936961196138621861869813045, 8.301597533202526663696302560632, 9.290724061478787067463164290929
