| L(s) = 1 | − 2-s + 4-s − 5·7-s − 8-s − 3·9-s + 2·11-s − 2·13-s + 5·14-s + 16-s − 6·17-s + 3·18-s + 7·19-s − 2·22-s − 6·23-s − 5·25-s + 2·26-s − 5·28-s − 3·29-s + 2·31-s − 32-s + 6·34-s − 3·36-s − 37-s − 7·38-s − 9·41-s + 12·43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.88·7-s − 0.353·8-s − 9-s + 0.603·11-s − 0.554·13-s + 1.33·14-s + 1/4·16-s − 1.45·17-s + 0.707·18-s + 1.60·19-s − 0.426·22-s − 1.25·23-s − 25-s + 0.392·26-s − 0.944·28-s − 0.557·29-s + 0.359·31-s − 0.176·32-s + 1.02·34-s − 1/2·36-s − 0.164·37-s − 1.13·38-s − 1.40·41-s + 1.82·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 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
−11.58213239794896313457390750967, −10.28761525294710471211767072126, −9.485403787654819352282231757213, −8.916866517552647291235515039702, −7.54266208602824924195749165375, −6.53835771973631692587002532310, −5.74382242989528346650352558559, −3.73621126104640023985723853467, −2.55661849848079940008326913553, 0,
2.55661849848079940008326913553, 3.73621126104640023985723853467, 5.74382242989528346650352558559, 6.53835771973631692587002532310, 7.54266208602824924195749165375, 8.916866517552647291235515039702, 9.485403787654819352282231757213, 10.28761525294710471211767072126, 11.58213239794896313457390750967
