| L(s) = 1 | + 2·7-s + 3·11-s − 5·13-s − 6·17-s + 4·19-s − 3·23-s − 2·31-s − 11·37-s + 6·41-s − 4·43-s + 3·47-s − 3·49-s + 12·53-s − 9·59-s + 11·61-s + 14·67-s + 15·71-s − 2·73-s + 6·77-s + 10·79-s + 6·89-s − 10·91-s + 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.904·11-s − 1.38·13-s − 1.45·17-s + 0.917·19-s − 0.625·23-s − 0.359·31-s − 1.80·37-s + 0.937·41-s − 0.609·43-s + 0.437·47-s − 3/7·49-s + 1.64·53-s − 1.17·59-s + 1.40·61-s + 1.71·67-s + 1.78·71-s − 0.234·73-s + 0.683·77-s + 1.12·79-s + 0.635·89-s − 1.04·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.967304352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.967304352\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.70025621054246, −15.74645482385843, −15.51848975677117, −14.67682114083661, −14.26936656157533, −13.87497580981207, −13.06987695473265, −12.41683324556231, −11.78425267198759, −11.50677185532998, −10.72726462888989, −10.10586081116995, −9.388014939549346, −8.952857499785875, −8.219000515583015, −7.535588352199786, −6.939668135287227, −6.407396532662791, −5.338284972958888, −4.964069040501875, −4.163360739242628, −3.497597506268794, −2.327560123972402, −1.871474240955524, −0.6427060612257972,
0.6427060612257972, 1.871474240955524, 2.327560123972402, 3.497597506268794, 4.163360739242628, 4.964069040501875, 5.338284972958888, 6.407396532662791, 6.939668135287227, 7.535588352199786, 8.219000515583015, 8.952857499785875, 9.388014939549346, 10.10586081116995, 10.72726462888989, 11.50677185532998, 11.78425267198759, 12.41683324556231, 13.06987695473265, 13.87497580981207, 14.26936656157533, 14.67682114083661, 15.51848975677117, 15.74645482385843, 16.70025621054246
