| L(s) = 1 | + 8i·3-s − 4i·7-s − 37·9-s − 12·11-s − 58i·13-s − 66i·17-s − 100·19-s + 32·21-s − 132i·23-s − 80i·27-s + 90·29-s − 152·31-s − 96i·33-s + 34i·37-s + 464·39-s + ⋯ |
| L(s) = 1 | + 1.53i·3-s − 0.215i·7-s − 1.37·9-s − 0.328·11-s − 1.23i·13-s − 0.941i·17-s − 1.20·19-s + 0.332·21-s − 1.19i·23-s − 0.570i·27-s + 0.576·29-s − 0.880·31-s − 0.506i·33-s + 0.151i·37-s + 1.90·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7892828498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7892828498\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 8iT - 27T^{2} \) |
| 7 | \( 1 + 4iT - 343T^{2} \) |
| 11 | \( 1 + 12T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 66iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 100T + 6.85e3T^{2} \) |
| 23 | \( 1 + 132iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 90T + 2.43e4T^{2} \) |
| 31 | \( 1 + 152T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 438T + 6.89e4T^{2} \) |
| 43 | \( 1 + 32iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 204iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 222iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 420T + 2.05e5T^{2} \) |
| 61 | \( 1 - 902T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 432T + 3.57e5T^{2} \) |
| 73 | \( 1 - 362iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 160T + 4.93e5T^{2} \) |
| 83 | \( 1 + 72iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 810T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.10e3iT - 9.12e5T^{2} \) |
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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
−10.40742612228477583123339837037, −10.11215882525853187921306998321, −8.931889719329597217922339508713, −8.220958256286813827480688066395, −6.86957754128808352492833102484, −5.52432384743826982638099992003, −4.76787080715042475526701244230, −3.75213779564222301094101297170, −2.65077880984899148337869881597, −0.25853637319379430745198509969,
1.46989980773140512980886815240, 2.32184008131005847734259956559, 3.95786270760744858547981600387, 5.50510706507821248227493189054, 6.50933563664974416561802883032, 7.13531374402991349644883425910, 8.193648389541321052519521268420, 8.864491989105973579984721724205, 10.15791095402599830935992690217, 11.32126578633781093636372224421
