L(s) = 1 | − 1.73·2-s + 0.999·4-s + 3.46·7-s + 1.73·8-s − 3.46·11-s − 5.99·14-s − 5·16-s − 6·17-s + 3.46·19-s + 5.99·22-s − 5·25-s + 3.46·28-s − 6·29-s − 3.46·31-s + 5.19·32-s + 10.3·34-s − 6.92·37-s − 5.99·38-s + 6.92·41-s + 4·43-s − 3.46·44-s + 3.46·47-s + 4.99·49-s + 8.66·50-s − 6·53-s + 6.00·56-s + 10.3·58-s + ⋯ |
L(s) = 1 | − 1.22·2-s + 0.499·4-s + 1.30·7-s + 0.612·8-s − 1.04·11-s − 1.60·14-s − 1.25·16-s − 1.45·17-s + 0.794·19-s + 1.27·22-s − 25-s + 0.654·28-s − 1.11·29-s − 0.622·31-s + 0.918·32-s + 1.78·34-s − 1.13·37-s − 0.973·38-s + 1.08·41-s + 0.609·43-s − 0.522·44-s + 0.505·47-s + 0.714·49-s + 1.22·50-s − 0.824·53-s + 0.801·56-s + 1.36·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.998764442342834100165097184971, −8.332257825284256696630759004412, −7.61922832362562448685817427313, −7.17482099423046659262209518890, −5.72983536965446822808509467013, −4.92051569999899627639582107947, −4.06264551255806293159961850586, −2.39455644807448585223645387954, −1.55463176831811738397772128768, 0,
1.55463176831811738397772128768, 2.39455644807448585223645387954, 4.06264551255806293159961850586, 4.92051569999899627639582107947, 5.72983536965446822808509467013, 7.17482099423046659262209518890, 7.61922832362562448685817427313, 8.332257825284256696630759004412, 8.998764442342834100165097184971