| L(s) = 1 | + 0.874·3-s − 2.23·9-s − 11-s + 2.28·13-s − 1.74·17-s − 3.16·19-s + 23-s − 4.57·27-s + 4.70·29-s + 6.86·31-s − 0.874·33-s − 4.23·37-s + 2·39-s − 0.206·41-s + 2.23·43-s − 1.41·47-s − 1.52·51-s + 3.23·53-s − 2.76·57-s + 1.62·59-s − 1.95·61-s − 1.47·67-s + 0.874·69-s − 6.70·71-s − 12.8·73-s + 5.94·79-s + 2.70·81-s + ⋯ |
| L(s) = 1 | + 0.504·3-s − 0.745·9-s − 0.301·11-s + 0.634·13-s − 0.423·17-s − 0.725·19-s + 0.208·23-s − 0.880·27-s + 0.874·29-s + 1.23·31-s − 0.152·33-s − 0.696·37-s + 0.320·39-s − 0.0322·41-s + 0.340·43-s − 0.206·47-s − 0.213·51-s + 0.444·53-s − 0.366·57-s + 0.210·59-s − 0.250·61-s − 0.179·67-s + 0.105·69-s − 0.796·71-s − 1.50·73-s + 0.668·79-s + 0.300·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.874T + 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 4.70T + 29T^{2} \) |
| 31 | \( 1 - 6.86T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 + 0.206T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 - 1.62T + 59T^{2} \) |
| 61 | \( 1 + 1.95T + 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 + 6.70T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 5.94T + 79T^{2} \) |
| 83 | \( 1 + 1.20T + 83T^{2} \) |
| 89 | \( 1 + 5.11T + 89T^{2} \) |
| 97 | \( 1 + 2.95T + 97T^{2} \) |
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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
−7.42823843928603281335795913692, −6.55374780915195488633566421484, −6.08929271734519760856495237581, −5.26805504511782616873915266734, −4.50259622043833041673650467347, −3.75319665529572588592484055812, −2.90453125945152679497049463070, −2.37429289353442788701379557178, −1.27502709884268367960738509428, 0,
1.27502709884268367960738509428, 2.37429289353442788701379557178, 2.90453125945152679497049463070, 3.75319665529572588592484055812, 4.50259622043833041673650467347, 5.26805504511782616873915266734, 6.08929271734519760856495237581, 6.55374780915195488633566421484, 7.42823843928603281335795913692
