L(s) = 1 | − 5-s + 7-s + 2·11-s + 4·13-s − 2·17-s + 2·19-s − 4·23-s + 25-s − 6·29-s − 2·31-s − 35-s + 10·37-s + 10·41-s + 12·43-s + 8·47-s + 49-s − 2·55-s + 8·59-s − 2·61-s − 4·65-s − 12·67-s + 10·71-s + 4·73-s + 2·77-s + 12·83-s + 2·85-s − 2·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.603·11-s + 1.10·13-s − 0.485·17-s + 0.458·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s − 0.169·35-s + 1.64·37-s + 1.56·41-s + 1.82·43-s + 1.16·47-s + 1/7·49-s − 0.269·55-s + 1.04·59-s − 0.256·61-s − 0.496·65-s − 1.46·67-s + 1.18·71-s + 0.468·73-s + 0.227·77-s + 1.31·83-s + 0.216·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.695840180\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.695840180\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.459268171549549227560669133895, −8.977489293175671900227133863971, −7.972522006250822658500010164715, −7.40282489558306515108943317960, −6.27575535344744739082506805421, −5.62102206398535181378106731263, −4.28120249162261891993830881046, −3.79614462524231983446903246185, −2.38904886593758690492350473788, −1.01812237326846089124898046997,
1.01812237326846089124898046997, 2.38904886593758690492350473788, 3.79614462524231983446903246185, 4.28120249162261891993830881046, 5.62102206398535181378106731263, 6.27575535344744739082506805421, 7.40282489558306515108943317960, 7.972522006250822658500010164715, 8.977489293175671900227133863971, 9.459268171549549227560669133895