| L(s) = 1 | + 1.08·2-s − 3-s − 0.827·4-s + 1.51·5-s − 1.08·6-s + 7-s − 3.06·8-s + 9-s + 1.64·10-s − 3.12·11-s + 0.827·12-s + 13-s + 1.08·14-s − 1.51·15-s − 1.66·16-s + 17-s + 1.08·18-s − 3.26·19-s − 1.25·20-s − 21-s − 3.38·22-s + 8.59·23-s + 3.06·24-s − 2.69·25-s + 1.08·26-s − 27-s − 0.827·28-s + ⋯ |
| L(s) = 1 | + 0.765·2-s − 0.577·3-s − 0.413·4-s + 0.679·5-s − 0.442·6-s + 0.377·7-s − 1.08·8-s + 0.333·9-s + 0.520·10-s − 0.943·11-s + 0.238·12-s + 0.277·13-s + 0.289·14-s − 0.392·15-s − 0.415·16-s + 0.242·17-s + 0.255·18-s − 0.749·19-s − 0.280·20-s − 0.218·21-s − 0.722·22-s + 1.79·23-s + 0.624·24-s − 0.538·25-s + 0.212·26-s − 0.192·27-s − 0.156·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.977619760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.977619760\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 1.08T + 2T^{2} \) |
| 5 | \( 1 - 1.51T + 5T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 8.59T + 23T^{2} \) |
| 29 | \( 1 - 3.19T + 29T^{2} \) |
| 31 | \( 1 - 1.60T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 0.978T + 41T^{2} \) |
| 43 | \( 1 - 0.615T + 43T^{2} \) |
| 47 | \( 1 - 5.52T + 47T^{2} \) |
| 53 | \( 1 - 5.08T + 53T^{2} \) |
| 59 | \( 1 + 0.222T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 2.10T + 71T^{2} \) |
| 73 | \( 1 + 7.61T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 0.180T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.502249900059831137578850052084, −7.42726778634338387547369930900, −6.63033557504884143185113497414, −5.88575001786267123942851904300, −5.22702183057759880806688333294, −4.88445851081185873485922510588, −3.92746707552607928357466790195, −3.01727391769732804751008712979, −2.04435080574755043077031443904, −0.71428981189975202419549183541,
0.71428981189975202419549183541, 2.04435080574755043077031443904, 3.01727391769732804751008712979, 3.92746707552607928357466790195, 4.88445851081185873485922510588, 5.22702183057759880806688333294, 5.88575001786267123942851904300, 6.63033557504884143185113497414, 7.42726778634338387547369930900, 8.502249900059831137578850052084
