| L(s) = 1 | + 1.73i·3-s + (−6.43 − 2.76i)7-s − 2.99·9-s − 20.9·11-s − 8.71i·13-s + 5.32i·17-s − 26.2i·19-s + (4.78 − 11.1i)21-s + 13.8·23-s − 5.19i·27-s + 32.3·29-s − 9.37i·31-s − 36.3i·33-s − 71.3·37-s + 15.0·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.918 − 0.394i)7-s − 0.333·9-s − 1.90·11-s − 0.670i·13-s + 0.313i·17-s − 1.38i·19-s + (0.227 − 0.530i)21-s + 0.601·23-s − 0.192i·27-s + 1.11·29-s − 0.302i·31-s − 1.10i·33-s − 1.92·37-s + 0.386·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9358460526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9358460526\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.43 + 2.76i)T \) |
| good | 11 | \( 1 + 20.9T + 121T^{2} \) |
| 13 | \( 1 + 8.71iT - 169T^{2} \) |
| 17 | \( 1 - 5.32iT - 289T^{2} \) |
| 19 | \( 1 + 26.2iT - 361T^{2} \) |
| 23 | \( 1 - 13.8T + 529T^{2} \) |
| 29 | \( 1 - 32.3T + 841T^{2} \) |
| 31 | \( 1 + 9.37iT - 961T^{2} \) |
| 37 | \( 1 + 71.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 54.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 34.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 30.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 63.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 98.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 68.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 34.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 12.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 122. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 47.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 153. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 139. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 40.8iT - 9.40e3T^{2} \) |
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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
−9.041085719468534966681837622725, −8.407220297192989626170075243852, −7.47423409880603309744459370758, −6.81150476376299094961906871094, −5.71620077567452656442627205788, −5.10874481384017148605740324151, −4.22361791897015224835889337246, −3.02545593552074346824941648562, −2.65604688629901922930257000097, −0.67438341591011763240914668347,
0.35203627808240386397759681640, 1.91809606396995076537662293444, 2.77747867637779721996580532439, 3.61135223718446651045180114836, 4.99318626919861118699189673941, 5.60580774965903869825343285166, 6.49366086635088346511304633897, 7.20102827878724729825891289040, 7.989574792709116776938359176915, 8.686669545143308295721091079680
