| L(s) = 1 | + 2-s − 4-s − 2·7-s − 3·8-s − 3·9-s − 4·11-s + 2·13-s − 2·14-s − 16-s − 4·17-s − 3·18-s + 19-s − 4·22-s + 6·23-s + 2·26-s + 2·28-s − 6·29-s − 4·31-s + 5·32-s − 4·34-s + 3·36-s + 10·37-s + 38-s − 10·41-s − 2·43-s + 4·44-s + 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.755·7-s − 1.06·8-s − 9-s − 1.20·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s − 0.970·17-s − 0.707·18-s + 0.229·19-s − 0.852·22-s + 1.25·23-s + 0.392·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.883·32-s − 0.685·34-s + 1/2·36-s + 1.64·37-s + 0.162·38-s − 1.56·41-s − 0.304·43-s + 0.603·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−10.75372555761590942712128571003, −9.497197103514413153205439327666, −8.884496394435335630734933815790, −7.910634270974764100222562199420, −6.56197008088421338178147885329, −5.66703630365227380430433843993, −4.88960236344040300877313120060, −3.56253018952603651968369317882, −2.71685356445346523746983648243, 0,
2.71685356445346523746983648243, 3.56253018952603651968369317882, 4.88960236344040300877313120060, 5.66703630365227380430433843993, 6.56197008088421338178147885329, 7.910634270974764100222562199420, 8.884496394435335630734933815790, 9.497197103514413153205439327666, 10.75372555761590942712128571003
