| L(s) = 1 | + 2·2-s + 8·3-s + 4·4-s − 14·5-s + 16·6-s + 7·7-s + 8·8-s + 37·9-s − 28·10-s + 32·12-s − 18·13-s + 14·14-s − 112·15-s + 16·16-s − 74·17-s + 74·18-s − 80·19-s − 56·20-s + 56·21-s − 112·23-s + 64·24-s + 71·25-s − 36·26-s + 80·27-s + 28·28-s − 190·29-s − 224·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s + 1/2·4-s − 1.25·5-s + 1.08·6-s + 0.377·7-s + 0.353·8-s + 1.37·9-s − 0.885·10-s + 0.769·12-s − 0.384·13-s + 0.267·14-s − 1.92·15-s + 1/4·16-s − 1.05·17-s + 0.968·18-s − 0.965·19-s − 0.626·20-s + 0.581·21-s − 1.01·23-s + 0.544·24-s + 0.567·25-s − 0.271·26-s + 0.570·27-s + 0.188·28-s − 1.21·29-s − 1.36·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 13 | \( 1 + 18 T + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 80 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 346 T + p^{3} T^{2} \) |
| 41 | \( 1 + 162 T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 24 T + p^{3} T^{2} \) |
| 53 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 198 T + p^{3} T^{2} \) |
| 67 | \( 1 + 716 T + p^{3} T^{2} \) |
| 71 | \( 1 - 392 T + p^{3} T^{2} \) |
| 73 | \( 1 + 538 T + p^{3} T^{2} \) |
| 79 | \( 1 + 240 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1072 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1354 T + p^{3} T^{2} \) |
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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
−8.555529669355491222011816634346, −7.69357401120284579981177003938, −7.34307839460776574593615123933, −6.26738532010575408193821147499, −4.92860824888504272064225674679, −4.03368112444339649976696614052, −3.72155626902725777716766715730, −2.56292220251372809249964421022, −1.85866396944695236054414895294, 0,
1.85866396944695236054414895294, 2.56292220251372809249964421022, 3.72155626902725777716766715730, 4.03368112444339649976696614052, 4.92860824888504272064225674679, 6.26738532010575408193821147499, 7.34307839460776574593615123933, 7.69357401120284579981177003938, 8.555529669355491222011816634346
