| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·11-s − 4.16·13-s − 14-s + 16-s + 7.32·17-s + 3·19-s − 2·22-s + 1.16·23-s − 4.16·26-s − 28-s − 1.83·29-s − 6.32·31-s + 32-s + 7.32·34-s − 7.48·37-s + 3·38-s − 4·41-s − 3.16·43-s − 2·44-s + 1.16·46-s − 10.4·47-s + 49-s − 4.16·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.377·7-s + 0.353·8-s − 0.603·11-s − 1.15·13-s − 0.267·14-s + 0.250·16-s + 1.77·17-s + 0.688·19-s − 0.426·22-s + 0.242·23-s − 0.816·26-s − 0.188·28-s − 0.341·29-s − 1.13·31-s + 0.176·32-s + 1.25·34-s − 1.23·37-s + 0.486·38-s − 0.624·41-s − 0.482·43-s − 0.301·44-s + 0.171·46-s − 1.52·47-s + 0.142·49-s − 0.577·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.16T + 13T^{2} \) |
| 17 | \( 1 - 7.32T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 + 1.83T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 0.162T + 53T^{2} \) |
| 59 | \( 1 - 0.324T + 59T^{2} \) |
| 61 | \( 1 + 3.83T + 61T^{2} \) |
| 67 | \( 1 - 3.48T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 4.32T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 + 5T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.33814831226958792545225417531, −6.71731811507376308906449755645, −5.76172556815274427737628196579, −5.24117678712513937042164438850, −4.82198878934548105443604222575, −3.55828587643474454239384058692, −3.28708533008611168687020028696, −2.35484546310722161372687020664, −1.39549316980368488449800425841, 0,
1.39549316980368488449800425841, 2.35484546310722161372687020664, 3.28708533008611168687020028696, 3.55828587643474454239384058692, 4.82198878934548105443604222575, 5.24117678712513937042164438850, 5.76172556815274427737628196579, 6.71731811507376308906449755645, 7.33814831226958792545225417531
