L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 5·5-s − 6·6-s − 8·8-s + 9·9-s − 10·10-s + 12·11-s + 12·12-s − 2·13-s + 15·15-s + 16·16-s + 18·17-s − 18·18-s − 56·19-s + 20·20-s − 24·22-s − 156·23-s − 24·24-s + 25·25-s + 4·26-s + 27·27-s − 186·29-s − 30·30-s + 52·31-s − 32·32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.328·11-s + 0.288·12-s − 0.0426·13-s + 0.258·15-s + 1/4·16-s + 0.256·17-s − 0.235·18-s − 0.676·19-s + 0.223·20-s − 0.232·22-s − 1.41·23-s − 0.204·24-s + 1/5·25-s + 0.0301·26-s + 0.192·27-s − 1.19·29-s − 0.182·30-s + 0.301·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + 156 T + p^{3} T^{2} \) |
| 29 | \( 1 + 186 T + p^{3} T^{2} \) |
| 31 | \( 1 - 52 T + p^{3} T^{2} \) |
| 37 | \( 1 + 178 T + p^{3} T^{2} \) |
| 41 | \( 1 - 138 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 456 T + p^{3} T^{2} \) |
| 53 | \( 1 + 198 T + p^{3} T^{2} \) |
| 59 | \( 1 + 348 T + p^{3} T^{2} \) |
| 61 | \( 1 + 110 T + p^{3} T^{2} \) |
| 67 | \( 1 + 196 T + p^{3} T^{2} \) |
| 71 | \( 1 + 936 T + p^{3} T^{2} \) |
| 73 | \( 1 + 542 T + p^{3} T^{2} \) |
| 79 | \( 1 - 992 T + p^{3} T^{2} \) |
| 83 | \( 1 - 276 T + p^{3} T^{2} \) |
| 89 | \( 1 + 630 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−8.813106417143478097037545597810, −8.048032859015243871711681595642, −7.31110220765731530170985723764, −6.39959106386420823447145973759, −5.63986886928639763555082630157, −4.36087778961802358995916971978, −3.38101008602024294046413920390, −2.25324618020060436337543677924, −1.48593178852078762408526314045, 0,
1.48593178852078762408526314045, 2.25324618020060436337543677924, 3.38101008602024294046413920390, 4.36087778961802358995916971978, 5.63986886928639763555082630157, 6.39959106386420823447145973759, 7.31110220765731530170985723764, 8.048032859015243871711681595642, 8.813106417143478097037545597810