| L(s) = 1 | − 3-s + 5-s − 2·9-s − 11-s + 5·13-s − 15-s − 17-s + 6·19-s − 4·23-s + 25-s + 5·27-s + 3·29-s − 2·31-s + 33-s + 8·37-s − 5·39-s + 10·41-s − 2·43-s − 2·45-s + 7·47-s + 51-s − 2·53-s − 55-s − 6·57-s − 14·59-s + 8·61-s + 5·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s − 0.301·11-s + 1.38·13-s − 0.258·15-s − 0.242·17-s + 1.37·19-s − 0.834·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 0.359·31-s + 0.174·33-s + 1.31·37-s − 0.800·39-s + 1.56·41-s − 0.304·43-s − 0.298·45-s + 1.02·47-s + 0.140·51-s − 0.274·53-s − 0.134·55-s − 0.794·57-s − 1.82·59-s + 1.02·61-s + 0.620·65-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
| \(L(1)\) |
\(\approx\) |
\(1.384395170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.384395170\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−10.03984630010892509244306051831, −9.205493091144591780558167948931, −8.374956661287084454927600986497, −7.50230990442548686845248936476, −6.24682000445599000021498906136, −5.87060629383830631719569211450, −4.92324231705683488838302429192, −3.68568172605374221523988395441, −2.54170572036124231642473845749, −0.982426883691529123696623033357,
0.982426883691529123696623033357, 2.54170572036124231642473845749, 3.68568172605374221523988395441, 4.92324231705683488838302429192, 5.87060629383830631719569211450, 6.24682000445599000021498906136, 7.50230990442548686845248936476, 8.374956661287084454927600986497, 9.205493091144591780558167948931, 10.03984630010892509244306051831
