L(s) = 1 | + 3-s + 5-s − 7-s − 2·9-s + 3·11-s + 15-s − 3·17-s − 7·19-s − 21-s − 3·23-s + 25-s − 5·27-s + 3·29-s − 4·31-s + 3·33-s − 35-s − 7·37-s − 9·41-s + 11·43-s − 2·45-s − 6·49-s − 3·51-s − 6·53-s + 3·55-s − 7·57-s − 3·59-s + 11·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.258·15-s − 0.727·17-s − 1.60·19-s − 0.218·21-s − 0.625·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s − 0.718·31-s + 0.522·33-s − 0.169·35-s − 1.15·37-s − 1.40·41-s + 1.67·43-s − 0.298·45-s − 6/7·49-s − 0.420·51-s − 0.824·53-s + 0.404·55-s − 0.927·57-s − 0.390·59-s + 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.573685428336991785362698123027, −7.54875420741161500047894722580, −6.51515890404852821085642253585, −6.25771160520714208967620241917, −5.21200756421472684184614643264, −4.20237940697639162223136467846, −3.48515912939368008496545886632, −2.47802326885889408042417662476, −1.73164036189747937634724332673, 0,
1.73164036189747937634724332673, 2.47802326885889408042417662476, 3.48515912939368008496545886632, 4.20237940697639162223136467846, 5.21200756421472684184614643264, 6.25771160520714208967620241917, 6.51515890404852821085642253585, 7.54875420741161500047894722580, 8.573685428336991785362698123027