| L(s) = 1 | − 2-s − 1.41·3-s + 4-s − 3.41·5-s + 1.41·6-s − 8-s − 0.999·9-s + 3.41·10-s + 4.24·11-s − 1.41·12-s − 0.828·13-s + 4.82·15-s + 16-s + 17-s + 0.999·18-s − 1.17·19-s − 3.41·20-s − 4.24·22-s + 2.82·23-s + 1.41·24-s + 6.65·25-s + 0.828·26-s + 5.65·27-s − 5.07·29-s − 4.82·30-s + 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.816·3-s + 0.5·4-s − 1.52·5-s + 0.577·6-s − 0.353·8-s − 0.333·9-s + 1.07·10-s + 1.27·11-s − 0.408·12-s − 0.229·13-s + 1.24·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.268·19-s − 0.763·20-s − 0.904·22-s + 0.589·23-s + 0.288·24-s + 1.33·25-s + 0.162·26-s + 1.08·27-s − 0.941·29-s − 0.881·30-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 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 5.07T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 7.17T + 47T^{2} \) |
| 53 | \( 1 + 8.82T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 + 1.17T + 71T^{2} \) |
| 73 | \( 1 + 6.48T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 + 9.65T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−8.897022544879507855637747593932, −8.171623087987009865258206611308, −7.43627601739549150469341002807, −6.66800778099619816535046584041, −5.93122127512132848618794303888, −4.75032891047545222162092595787, −3.91850568444440549461938381791, −2.90529638280549278013725850801, −1.15813368021054032941525813740, 0,
1.15813368021054032941525813740, 2.90529638280549278013725850801, 3.91850568444440549461938381791, 4.75032891047545222162092595787, 5.93122127512132848618794303888, 6.66800778099619816535046584041, 7.43627601739549150469341002807, 8.171623087987009865258206611308, 8.897022544879507855637747593932
