L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s − 13-s + 16-s + 19-s − 4·22-s − 4·23-s − 26-s + 3·29-s + 4·31-s + 32-s − 5·37-s + 38-s − 9·41-s + 2·43-s − 4·44-s − 4·46-s + 3·47-s − 7·49-s − 52-s − 53-s + 3·58-s − 10·59-s + 4·61-s + 4·62-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s − 0.277·13-s + 1/4·16-s + 0.229·19-s − 0.852·22-s − 0.834·23-s − 0.196·26-s + 0.557·29-s + 0.718·31-s + 0.176·32-s − 0.821·37-s + 0.162·38-s − 1.40·41-s + 0.304·43-s − 0.603·44-s − 0.589·46-s + 0.437·47-s − 49-s − 0.138·52-s − 0.137·53-s + 0.393·58-s − 1.30·59-s + 0.512·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.78347466719420967839024436887, −6.93453351459652488383385266697, −6.27008192303997909483107254830, −5.44062083512289591280529258486, −4.92560921649166282848960041209, −4.14525765885028334219702874665, −3.18093490140462612166508283981, −2.55408633874181044021542323396, −1.55154893905601864682168759515, 0,
1.55154893905601864682168759515, 2.55408633874181044021542323396, 3.18093490140462612166508283981, 4.14525765885028334219702874665, 4.92560921649166282848960041209, 5.44062083512289591280529258486, 6.27008192303997909483107254830, 6.93453351459652488383385266697, 7.78347466719420967839024436887