| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 5·13-s − 14-s + 16-s + 3·17-s + 2·19-s − 9·23-s − 5·26-s − 28-s + 3·29-s + 5·31-s + 32-s + 3·34-s − 2·37-s + 2·38-s + 6·41-s + 43-s − 9·46-s − 6·47-s + 49-s − 5·52-s + 3·53-s − 56-s + 3·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.38·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s − 1.87·23-s − 0.980·26-s − 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s + 0.514·34-s − 0.328·37-s + 0.324·38-s + 0.937·41-s + 0.152·43-s − 1.32·46-s − 0.875·47-s + 1/7·49-s − 0.693·52-s + 0.412·53-s − 0.133·56-s + 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.29199281546736967538845772277, −6.62139626687353922215576455203, −5.87287819102938113469707399631, −5.34622096867902893048908248821, −4.50987208135388742371514439705, −3.94841862875190945765095501059, −2.97249796179752463562060054454, −2.46273168262964029176117899456, −1.38076542358485290966418709809, 0,
1.38076542358485290966418709809, 2.46273168262964029176117899456, 2.97249796179752463562060054454, 3.94841862875190945765095501059, 4.50987208135388742371514439705, 5.34622096867902893048908248821, 5.87287819102938113469707399631, 6.62139626687353922215576455203, 7.29199281546736967538845772277
