| L(s) = 1 | − 1.56·3-s − 5-s + 7-s − 0.561·9-s − 1.56·11-s − 3.56·13-s + 1.56·15-s + 0.438·17-s − 1.56·21-s + 25-s + 5.56·27-s − 6.68·29-s + 3.12·31-s + 2.43·33-s − 35-s + 1.12·37-s + 5.56·39-s + 2·41-s + 4·43-s + 0.561·45-s − 0.684·47-s + 49-s − 0.684·51-s − 13.1·53-s + 1.56·55-s + 6.24·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.901·3-s − 0.447·5-s + 0.377·7-s − 0.187·9-s − 0.470·11-s − 0.987·13-s + 0.403·15-s + 0.106·17-s − 0.340·21-s + 0.200·25-s + 1.07·27-s − 1.24·29-s + 0.560·31-s + 0.424·33-s − 0.169·35-s + 0.184·37-s + 0.890·39-s + 0.312·41-s + 0.609·43-s + 0.0837·45-s − 0.0998·47-s + 0.142·49-s − 0.0958·51-s − 1.80·53-s + 0.210·55-s + 0.813·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7946018049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7946018049\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 0.684T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 0.246T + 73T^{2} \) |
| 79 | \( 1 + 7.80T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−9.049821303742649391373595156912, −8.096262004580016505674474027318, −7.53460299832476471237283423719, −6.67264519206290839108154027445, −5.77403380216881752251364300192, −5.10118796806901444798421924921, −4.42531352134838379430142830981, −3.23643208297523430129350864074, −2.15584734182179304000462146284, −0.58526933706795397978733388340,
0.58526933706795397978733388340, 2.15584734182179304000462146284, 3.23643208297523430129350864074, 4.42531352134838379430142830981, 5.10118796806901444798421924921, 5.77403380216881752251364300192, 6.67264519206290839108154027445, 7.53460299832476471237283423719, 8.096262004580016505674474027318, 9.049821303742649391373595156912
