| L(s) = 1 | − 7·3-s − 5·5-s − 11·7-s + 22·9-s − 11·11-s + 2·13-s + 35·15-s − 9·17-s + 85·19-s + 77·21-s + 138·23-s + 25·25-s + 35·27-s + 45·29-s − 227·31-s + 77·33-s + 55·35-s − 19·37-s − 14·39-s − 138·41-s + 88·43-s − 110·45-s + 534·47-s − 222·49-s + 63·51-s + 297·53-s + 55·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s − 0.447·5-s − 0.593·7-s + 0.814·9-s − 0.301·11-s + 0.0426·13-s + 0.602·15-s − 0.128·17-s + 1.02·19-s + 0.800·21-s + 1.25·23-s + 1/5·25-s + 0.249·27-s + 0.288·29-s − 1.31·31-s + 0.406·33-s + 0.265·35-s − 0.0844·37-s − 0.0574·39-s − 0.525·41-s + 0.312·43-s − 0.364·45-s + 1.65·47-s − 0.647·49-s + 0.172·51-s + 0.769·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 11 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 9 T + p^{3} T^{2} \) |
| 19 | \( 1 - 85 T + p^{3} T^{2} \) |
| 23 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 45 T + p^{3} T^{2} \) |
| 31 | \( 1 + 227 T + p^{3} T^{2} \) |
| 37 | \( 1 + 19 T + p^{3} T^{2} \) |
| 41 | \( 1 + 138 T + p^{3} T^{2} \) |
| 43 | \( 1 - 88 T + p^{3} T^{2} \) |
| 47 | \( 1 - 534 T + p^{3} T^{2} \) |
| 53 | \( 1 - 297 T + p^{3} T^{2} \) |
| 59 | \( 1 - 450 T + p^{3} T^{2} \) |
| 61 | \( 1 - 287 T + p^{3} T^{2} \) |
| 67 | \( 1 - 304 T + p^{3} T^{2} \) |
| 71 | \( 1 + 777 T + p^{3} T^{2} \) |
| 73 | \( 1 - 962 T + p^{3} T^{2} \) |
| 79 | \( 1 + 290 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1422 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1455 T + p^{3} T^{2} \) |
| 97 | \( 1 - 116 T + p^{3} T^{2} \) |
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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
−9.470264192878884417116399756639, −8.528281750442619553156266314627, −7.28759011015293977137901958867, −6.79527086854771605046572022606, −5.66514768386910038342425093425, −5.15954451612078368867919757686, −3.98348805585045624980963960020, −2.84765454462514917146403668171, −1.04948356362849959783423591545, 0,
1.04948356362849959783423591545, 2.84765454462514917146403668171, 3.98348805585045624980963960020, 5.15954451612078368867919757686, 5.66514768386910038342425093425, 6.79527086854771605046572022606, 7.28759011015293977137901958867, 8.528281750442619553156266314627, 9.470264192878884417116399756639
