| L(s) = 1 | + 2-s − 2·3-s + 4-s − 5-s − 2·6-s + 8-s + 9-s − 10-s + 3·11-s − 2·12-s + 5·13-s + 2·15-s + 16-s − 6·17-s + 18-s − 20-s + 3·22-s − 2·24-s + 25-s + 5·26-s + 4·27-s − 6·29-s + 2·30-s − 4·31-s + 32-s − 6·33-s − 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s − 0.577·12-s + 1.38·13-s + 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.223·20-s + 0.639·22-s − 0.408·24-s + 1/5·25-s + 0.980·26-s + 0.769·27-s − 1.11·29-s + 0.365·30-s − 0.718·31-s + 0.176·32-s − 1.04·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9405307698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9405307698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.08884010122111, −12.82072083445169, −12.00811331056798, −11.79199491141083, −11.30725913269417, −11.06784929668741, −10.65066049826249, −10.10236688598236, −9.261212140962796, −8.888154742035583, −8.452299078380174, −7.771393003329600, −7.108732725056109, −6.649559401758003, −6.361736874986787, −5.825379947259668, −5.380142964294578, −4.787220357791162, −4.192409789963848, −3.883519895814750, −3.287097015620039, −2.581255500966435, −1.624501114701151, −1.332777376585259, −0.2679677290085607,
0.2679677290085607, 1.332777376585259, 1.624501114701151, 2.581255500966435, 3.287097015620039, 3.883519895814750, 4.192409789963848, 4.787220357791162, 5.380142964294578, 5.825379947259668, 6.361736874986787, 6.649559401758003, 7.108732725056109, 7.771393003329600, 8.452299078380174, 8.888154742035583, 9.261212140962796, 10.10236688598236, 10.65066049826249, 11.06784929668741, 11.30725913269417, 11.79199491141083, 12.00811331056798, 12.82072083445169, 13.08884010122111
