| L(s) = 1 | + 2·7-s + 4·11-s + 4·13-s − 4·19-s + 2·23-s − 2·29-s + 4·37-s − 2·41-s − 6·43-s + 6·47-s − 3·49-s + 4·53-s + 12·59-s − 10·61-s + 14·67-s − 8·71-s + 8·73-s + 8·77-s + 16·79-s − 2·83-s − 6·89-s + 8·91-s + 16·97-s − 6·101-s + 14·103-s + 10·107-s − 6·109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.20·11-s + 1.10·13-s − 0.917·19-s + 0.417·23-s − 0.371·29-s + 0.657·37-s − 0.312·41-s − 0.914·43-s + 0.875·47-s − 3/7·49-s + 0.549·53-s + 1.56·59-s − 1.28·61-s + 1.71·67-s − 0.949·71-s + 0.936·73-s + 0.911·77-s + 1.80·79-s − 0.219·83-s − 0.635·89-s + 0.838·91-s + 1.62·97-s − 0.597·101-s + 1.37·103-s + 0.966·107-s − 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.138817306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.138817306\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.076671008574562961831249636879, −8.585728195214861045675483029522, −7.81135491243190146866399938297, −6.76652613570959587126287493943, −6.19373330497638433979341785295, −5.18231490639048164598324241536, −4.22149194734832020050341258471, −3.53324038675806191397269745952, −2.09779734649908396191483201131, −1.09129333142894422564489881503,
1.09129333142894422564489881503, 2.09779734649908396191483201131, 3.53324038675806191397269745952, 4.22149194734832020050341258471, 5.18231490639048164598324241536, 6.19373330497638433979341785295, 6.76652613570959587126287493943, 7.81135491243190146866399938297, 8.585728195214861045675483029522, 9.076671008574562961831249636879
