| L(s) = 1 | + 2-s − 3·3-s + 4-s − 2·5-s − 3·6-s + 7-s + 8-s + 6·9-s − 2·10-s − 11-s − 3·12-s + 14-s + 6·15-s + 16-s + 17-s + 6·18-s + 6·19-s − 2·20-s − 3·21-s − 22-s − 2·23-s − 3·24-s − 25-s − 9·27-s + 28-s − 4·29-s + 6·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.894·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s − 0.632·10-s − 0.301·11-s − 0.866·12-s + 0.267·14-s + 1.54·15-s + 1/4·16-s + 0.242·17-s + 1.41·18-s + 1.37·19-s − 0.447·20-s − 0.654·21-s − 0.213·22-s − 0.417·23-s − 0.612·24-s − 1/5·25-s − 1.73·27-s + 0.188·28-s − 0.742·29-s + 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.65714926953908, −15.33267593331260, −14.39399419256758, −14.09655194760862, −13.26082937901425, −12.61668398403617, −12.38256191214156, −11.72041221845525, −11.34524065808063, −11.08834395884868, −10.42905266401812, −9.786349320757090, −9.193187644148849, −7.966337412039215, −7.719401256227371, −7.172738138943048, −6.444087518816733, −5.874129574263816, −5.340985175748807, −4.904660516928960, −4.206480929581695, −3.724179149641808, −2.837144745709186, −1.716333500739265, −0.9239395263766036, 0,
0.9239395263766036, 1.716333500739265, 2.837144745709186, 3.724179149641808, 4.206480929581695, 4.904660516928960, 5.340985175748807, 5.874129574263816, 6.444087518816733, 7.172738138943048, 7.719401256227371, 7.966337412039215, 9.193187644148849, 9.786349320757090, 10.42905266401812, 11.08834395884868, 11.34524065808063, 11.72041221845525, 12.38256191214156, 12.61668398403617, 13.26082937901425, 14.09655194760862, 14.39399419256758, 15.33267593331260, 15.65714926953908
