| L(s) = 1 | − 1.41·2-s + 1.73i·3-s + 2.00·4-s − 2.44i·6-s + (−0.853 − 6.94i)7-s − 2.82·8-s − 2.99·9-s + 2.88·11-s + 3.46i·12-s + 13.8i·13-s + (1.20 + 9.82i)14-s + 4.00·16-s − 24.1i·17-s + 4.24·18-s + 6.53i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (−0.121 − 0.992i)7-s − 0.353·8-s − 0.333·9-s + 0.261·11-s + 0.288i·12-s + 1.06i·13-s + (0.0861 + 0.701i)14-s + 0.250·16-s − 1.42i·17-s + 0.235·18-s + 0.343i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06219265009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06219265009\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.853 + 6.94i)T \) |
| good | 11 | \( 1 - 2.88T + 121T^{2} \) |
| 13 | \( 1 - 13.8iT - 169T^{2} \) |
| 17 | \( 1 + 24.1iT - 289T^{2} \) |
| 19 | \( 1 - 6.53iT - 361T^{2} \) |
| 23 | \( 1 + 28.8T + 529T^{2} \) |
| 29 | \( 1 - 32.9T + 841T^{2} \) |
| 31 | \( 1 - 2.43iT - 961T^{2} \) |
| 37 | \( 1 + 50.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 21.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 13.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 40.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 17.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 1.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 120.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 21.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 66.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 78.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 90.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 44.1iT - 9.40e3T^{2} \) |
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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
−10.13184331326161985727743673335, −9.386722274455433754666880445885, −8.687242682159363597856495345997, −7.65833481848891307093292186757, −6.94314085782451910912803853649, −6.10783962869203855712609823630, −4.78644694695130524402931260379, −4.00126976685593565538081729647, −2.86593908604901864543173488934, −1.42650306753293439888644360054,
0.02436939212777101188929617006, 1.55068958920343672766935492373, 2.53810416408907629396355850991, 3.65085139959642129282640039341, 5.25532081635644172602901003876, 6.07086128768614752957107299013, 6.73150053990607678148570910137, 7.931514941288247765245000033738, 8.370314627608094957266514820306, 9.111617404566418764203388197412
