| L(s) = 1 | + 2·2-s − 7.17·3-s + 4·4-s − 18.7·5-s − 14.3·6-s + 8·8-s + 24.5·9-s − 37.4·10-s − 36.8·11-s − 28.7·12-s − 22.7·13-s + 134.·15-s + 16·16-s − 66.6·17-s + 49.0·18-s + 106.·19-s − 74.8·20-s − 73.7·22-s + 23·23-s − 57.4·24-s + 224.·25-s − 45.5·26-s + 17.8·27-s − 199.·29-s + 268.·30-s + 262.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.38·3-s + 0.5·4-s − 1.67·5-s − 0.976·6-s + 0.353·8-s + 0.907·9-s − 1.18·10-s − 1.01·11-s − 0.690·12-s − 0.486·13-s + 2.31·15-s + 0.250·16-s − 0.951·17-s + 0.641·18-s + 1.28·19-s − 0.836·20-s − 0.715·22-s + 0.208·23-s − 0.488·24-s + 1.79·25-s − 0.343·26-s + 0.127·27-s − 1.27·29-s + 1.63·30-s + 1.51·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 7.17T + 27T^{2} \) |
| 5 | \( 1 + 18.7T + 125T^{2} \) |
| 11 | \( 1 + 36.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 66.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 106.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 199.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 262.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 44.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 18.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 255.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 598.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 636.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 320.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 24.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 767.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 807.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
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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
−7.81668494591790018903695330043, −7.50722505410815247527426742747, −6.65409063275538606004758090844, −5.79175468821126784880144600066, −4.92195948375850964002006743786, −4.55500603692004924480532023663, −3.54063398001694768403749582154, −2.57879205759713552607542495647, −0.846043750499627901839896537967, 0,
0.846043750499627901839896537967, 2.57879205759713552607542495647, 3.54063398001694768403749582154, 4.55500603692004924480532023663, 4.92195948375850964002006743786, 5.79175468821126784880144600066, 6.65409063275538606004758090844, 7.50722505410815247527426742747, 7.81668494591790018903695330043
