| L(s) = 1 | − 3·3-s + 7-s + 6·9-s − 5·11-s + 3·13-s + 17-s + 6·19-s − 3·21-s − 6·23-s − 9·27-s − 9·29-s − 4·31-s + 15·33-s − 2·37-s − 9·39-s − 4·41-s − 10·43-s + 47-s + 49-s − 3·51-s − 4·53-s − 18·57-s − 8·59-s − 8·61-s + 6·63-s − 12·67-s + 18·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.377·7-s + 2·9-s − 1.50·11-s + 0.832·13-s + 0.242·17-s + 1.37·19-s − 0.654·21-s − 1.25·23-s − 1.73·27-s − 1.67·29-s − 0.718·31-s + 2.61·33-s − 0.328·37-s − 1.44·39-s − 0.624·41-s − 1.52·43-s + 0.145·47-s + 1/7·49-s − 0.420·51-s − 0.549·53-s − 2.38·57-s − 1.04·59-s − 1.02·61-s + 0.755·63-s − 1.46·67-s + 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−10.36920587717885305081018636748, −9.433095776999910916649205963026, −7.994283163097885160492521092184, −7.36618684826516234411138068520, −6.17794263542043593877652385699, −5.47599229122945418619459699774, −4.90541623417304819762465937335, −3.54576336197442309534683647891, −1.63989097462119583341292673385, 0,
1.63989097462119583341292673385, 3.54576336197442309534683647891, 4.90541623417304819762465937335, 5.47599229122945418619459699774, 6.17794263542043593877652385699, 7.36618684826516234411138068520, 7.994283163097885160492521092184, 9.433095776999910916649205963026, 10.36920587717885305081018636748
