| L(s) = 1 | − 3·7-s − 3·9-s + 11-s − 13-s + 2·17-s − 2·19-s + 3·23-s − 8·29-s + 2·37-s − 3·41-s + 8·43-s − 4·47-s + 2·49-s + 10·53-s − 9·59-s + 61-s + 9·63-s + 13·67-s + 12·71-s − 5·73-s − 3·77-s + 17·79-s + 9·81-s + 12·83-s − 8·89-s + 3·91-s + 18·97-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 9-s + 0.301·11-s − 0.277·13-s + 0.485·17-s − 0.458·19-s + 0.625·23-s − 1.48·29-s + 0.328·37-s − 0.468·41-s + 1.21·43-s − 0.583·47-s + 2/7·49-s + 1.37·53-s − 1.17·59-s + 0.128·61-s + 1.13·63-s + 1.58·67-s + 1.42·71-s − 0.585·73-s − 0.341·77-s + 1.91·79-s + 81-s + 1.31·83-s − 0.847·89-s + 0.314·91-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.63164132401286, −15.04589260484977, −14.68633367855619, −14.04714563217721, −13.51930314649429, −12.93521252021122, −12.47833655395207, −11.96642686761890, −11.24114493400158, −10.88862187482471, −10.15383761614502, −9.484962818893443, −9.215971197625754, −8.562017655072466, −7.857651738764556, −7.291306091272654, −6.532009683001533, −6.164692006389959, −5.453097733774525, −4.934501812994239, −3.836497092617163, −3.511669298290187, −2.711330779360823, −2.099747206279848, −0.8762754064350487, 0,
0.8762754064350487, 2.099747206279848, 2.711330779360823, 3.511669298290187, 3.836497092617163, 4.934501812994239, 5.453097733774525, 6.164692006389959, 6.532009683001533, 7.291306091272654, 7.857651738764556, 8.562017655072466, 9.215971197625754, 9.484962818893443, 10.15383761614502, 10.88862187482471, 11.24114493400158, 11.96642686761890, 12.47833655395207, 12.93521252021122, 13.51930314649429, 14.04714563217721, 14.68633367855619, 15.04589260484977, 15.63164132401286
