| L(s) = 1 | − 3-s − 7-s + 9-s − 2·13-s − 2·17-s + 4·19-s + 21-s − 27-s + 2·29-s + 4·31-s − 2·37-s + 2·39-s − 2·41-s − 8·43-s + 4·47-s + 49-s + 2·51-s + 2·53-s − 4·57-s + 12·59-s − 14·61-s − 63-s − 8·67-s − 8·71-s + 2·73-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.328·37-s + 0.320·39-s − 0.312·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.529·57-s + 1.56·59-s − 1.79·61-s − 0.125·63-s − 0.977·67-s − 0.949·71-s + 0.234·73-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−7.973540488422397924156745228235, −7.16288589339825161868250569411, −6.62931250253689451122375068463, −5.81102308344872542033025387528, −5.07986281807811606612952338917, −4.37705584426970065597223209755, −3.38134301278279514914593760545, −2.49120678334683844767562612178, −1.27440178887789443572179075537, 0,
1.27440178887789443572179075537, 2.49120678334683844767562612178, 3.38134301278279514914593760545, 4.37705584426970065597223209755, 5.07986281807811606612952338917, 5.81102308344872542033025387528, 6.62931250253689451122375068463, 7.16288589339825161868250569411, 7.973540488422397924156745228235
