L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s + 2·10-s + 4·11-s + 4·13-s + 14-s + 16-s + 8·17-s − 2·19-s + 2·20-s + 4·22-s − 23-s − 25-s + 4·26-s + 28-s − 2·29-s − 6·31-s + 32-s + 8·34-s + 2·35-s − 10·37-s − 2·38-s + 2·40-s − 6·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s + 0.632·10-s + 1.20·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s − 0.458·19-s + 0.447·20-s + 0.852·22-s − 0.208·23-s − 1/5·25-s + 0.784·26-s + 0.188·28-s − 0.371·29-s − 1.07·31-s + 0.176·32-s + 1.37·34-s + 0.338·35-s − 1.64·37-s − 0.324·38-s + 0.316·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.210706939\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.210706939\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.692325079374886601900730130410, −8.059585303969141042310152900127, −6.98268863135979537740471510303, −6.37674512300381969971763179171, −5.59998054730535419727722836986, −5.11526771523389587782932194500, −3.76192426912516843776743354862, −3.47343278151125646137638285313, −1.92837567429467993022158989891, −1.35429662567989955222011676765,
1.35429662567989955222011676765, 1.92837567429467993022158989891, 3.47343278151125646137638285313, 3.76192426912516843776743354862, 5.11526771523389587782932194500, 5.59998054730535419727722836986, 6.37674512300381969971763179171, 6.98268863135979537740471510303, 8.059585303969141042310152900127, 8.692325079374886601900730130410