| L(s) = 1 | − 3-s − 4·5-s − 7-s + 9-s + 2·11-s + 6·13-s + 4·15-s + 4·17-s + 4·19-s + 21-s − 2·23-s + 11·25-s − 27-s + 2·29-s − 2·33-s + 4·35-s − 6·39-s + 4·43-s − 4·45-s + 12·47-s + 49-s − 4·51-s − 6·53-s − 8·55-s − 4·57-s + 8·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.66·13-s + 1.03·15-s + 0.970·17-s + 0.917·19-s + 0.218·21-s − 0.417·23-s + 11/5·25-s − 0.192·27-s + 0.371·29-s − 0.348·33-s + 0.676·35-s − 0.960·39-s + 0.609·43-s − 0.596·45-s + 1.75·47-s + 1/7·49-s − 0.560·51-s − 0.824·53-s − 1.07·55-s − 0.529·57-s + 1.04·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114996 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114996 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.918461167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.918461167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.67598414499932, −12.82810077821521, −12.63966377149032, −11.96808996365427, −11.68376702131532, −11.37629493838592, −10.76886050394335, −10.38794550190391, −9.757033603955624, −9.027477717695618, −8.699215188791006, −8.104124450400553, −7.552269444908521, −7.285649551142118, −6.580128666768112, −6.071228385733783, −5.614231203616287, −4.888346951681059, −4.175566013384266, −3.906643759074441, −3.344594647228143, −2.932596568510471, −1.690222098201461, −0.8934775124011173, −0.6225691710175066,
0.6225691710175066, 0.8934775124011173, 1.690222098201461, 2.932596568510471, 3.344594647228143, 3.906643759074441, 4.175566013384266, 4.888346951681059, 5.614231203616287, 6.071228385733783, 6.580128666768112, 7.285649551142118, 7.552269444908521, 8.104124450400553, 8.699215188791006, 9.027477717695618, 9.757033603955624, 10.38794550190391, 10.76886050394335, 11.37629493838592, 11.68376702131532, 11.96808996365427, 12.63966377149032, 12.82810077821521, 13.67598414499932
