| L(s) = 1 | + 2·7-s + 11-s − 4·13-s + 5·17-s − 19-s + 2·23-s + 8·29-s − 10·31-s + 6·37-s + 3·41-s + 4·43-s − 4·47-s − 3·49-s + 6·53-s + 8·59-s + 10·61-s − 67-s − 12·71-s − 3·73-s + 2·77-s − 6·79-s + 13·83-s + 9·89-s − 8·91-s + 14·97-s − 6·101-s + 4·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s − 1.10·13-s + 1.21·17-s − 0.229·19-s + 0.417·23-s + 1.48·29-s − 1.79·31-s + 0.986·37-s + 0.468·41-s + 0.609·43-s − 0.583·47-s − 3/7·49-s + 0.824·53-s + 1.04·59-s + 1.28·61-s − 0.122·67-s − 1.42·71-s − 0.351·73-s + 0.227·77-s − 0.675·79-s + 1.42·83-s + 0.953·89-s − 0.838·91-s + 1.42·97-s − 0.597·101-s + 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.087822740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.087822740\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.531981611993951052357691184258, −7.69745703500215619459094697790, −7.28228166978740602694514888902, −6.31100545969129210351726422602, −5.41547212317211981015194114430, −4.83269960324324040804784090100, −3.96775721542160486783344312585, −2.94994957636342824563203996374, −2.00382939643555619997461574560, −0.871905856173456100249657871407,
0.871905856173456100249657871407, 2.00382939643555619997461574560, 2.94994957636342824563203996374, 3.96775721542160486783344312585, 4.83269960324324040804784090100, 5.41547212317211981015194114430, 6.31100545969129210351726422602, 7.28228166978740602694514888902, 7.69745703500215619459094697790, 8.531981611993951052357691184258
