| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s − 7-s − 2·9-s + 3·11-s − 2·12-s − 13-s − 2·14-s − 4·16-s − 7·17-s − 4·18-s + 21-s + 6·22-s − 6·23-s − 2·26-s + 5·27-s − 2·28-s + 5·29-s − 8·32-s − 3·33-s − 14·34-s − 4·36-s − 2·37-s + 39-s + 2·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s − 0.277·13-s − 0.534·14-s − 16-s − 1.69·17-s − 0.942·18-s + 0.218·21-s + 1.27·22-s − 1.25·23-s − 0.392·26-s + 0.962·27-s − 0.377·28-s + 0.928·29-s − 1.41·32-s − 0.522·33-s − 2.40·34-s − 2/3·36-s − 0.328·37-s + 0.160·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.46116018916633, −13.03175738654772, −12.48520446588452, −12.11609264427755, −11.66462603295253, −11.39190961340680, −10.85154207316904, −10.32657497479134, −9.655653298510648, −9.185681394250124, −8.633897511573177, −8.298388837164532, −7.383351271980296, −6.787701770092888, −6.432170695686167, −6.135686450787227, −5.577152203402406, −5.037668883453409, −4.475775364333045, −4.125946217946944, −3.566868563902065, −2.900190651857540, −2.384022188004677, −1.821773259014984, −0.6986669036867241, 0,
0.6986669036867241, 1.821773259014984, 2.384022188004677, 2.900190651857540, 3.566868563902065, 4.125946217946944, 4.475775364333045, 5.037668883453409, 5.577152203402406, 6.135686450787227, 6.432170695686167, 6.787701770092888, 7.383351271980296, 8.298388837164532, 8.633897511573177, 9.185681394250124, 9.655653298510648, 10.32657497479134, 10.85154207316904, 11.39190961340680, 11.66462603295253, 12.11609264427755, 12.48520446588452, 13.03175738654772, 13.46116018916633
