| L(s) = 1 | + 2·3-s + 5-s + 9-s − 2·13-s + 2·15-s + 6·17-s + 4·19-s + 6·23-s + 25-s − 4·27-s + 6·29-s + 4·31-s + 2·37-s − 4·39-s − 6·41-s − 10·43-s + 45-s + 6·47-s + 12·51-s − 6·53-s + 8·57-s − 12·59-s − 2·61-s − 2·65-s + 2·67-s + 12·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.640·39-s − 0.937·41-s − 1.52·43-s + 0.149·45-s + 0.875·47-s + 1.68·51-s − 0.824·53-s + 1.05·57-s − 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.244·67-s + 1.44·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.578660952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.578660952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.789054880187459544551674749578, −9.237629786112200017815782266993, −8.321958701563865500343115449553, −7.67946935430715687873464278567, −6.77106114591073562021381363331, −5.60226623689013744024577168227, −4.73531593841690414562612515004, −3.28633294922749820286908619224, −2.80176924686893785247315174730, −1.37767620904697515455020569610,
1.37767620904697515455020569610, 2.80176924686893785247315174730, 3.28633294922749820286908619224, 4.73531593841690414562612515004, 5.60226623689013744024577168227, 6.77106114591073562021381363331, 7.67946935430715687873464278567, 8.321958701563865500343115449553, 9.237629786112200017815782266993, 9.789054880187459544551674749578
