| L(s) = 1 | − 2-s + 1.41·3-s + 4-s − 0.585·5-s − 1.41·6-s − 8-s − 0.999·9-s + 0.585·10-s − 4.24·11-s + 1.41·12-s + 4.82·13-s − 0.828·15-s + 16-s + 17-s + 0.999·18-s − 6.82·19-s − 0.585·20-s + 4.24·22-s − 2.82·23-s − 1.41·24-s − 4.65·25-s − 4.82·26-s − 5.65·27-s + 9.07·29-s + 0.828·30-s + 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.816·3-s + 0.5·4-s − 0.261·5-s − 0.577·6-s − 0.353·8-s − 0.333·9-s + 0.185·10-s − 1.27·11-s + 0.408·12-s + 1.33·13-s − 0.213·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 1.56·19-s − 0.130·20-s + 0.904·22-s − 0.589·23-s − 0.288·24-s − 0.931·25-s − 0.946·26-s − 1.08·27-s + 1.68·29-s + 0.151·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 0.585T + 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 9.07T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 2.24T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 + 6.82T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 4.48T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.606143479438815997183461840597, −8.270041016200715636441473148381, −7.898596748882413482862852144252, −6.58691245789484462170700652837, −5.98086918592629650002333177224, −4.74433916485160356821846534194, −3.57093682085767875920397672063, −2.77716706695025822612191096366, −1.76144869243298086070313201817, 0,
1.76144869243298086070313201817, 2.77716706695025822612191096366, 3.57093682085767875920397672063, 4.74433916485160356821846534194, 5.98086918592629650002333177224, 6.58691245789484462170700652837, 7.898596748882413482862852144252, 8.270041016200715636441473148381, 8.606143479438815997183461840597
