| 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 + 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 + 10·41-s − 2·43-s + 4·44-s − 6·46-s + 6·47-s − 3·49-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 + 0.852·22-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 + 1.56·41-s − 0.304·43-s + 0.603·44-s − 0.884·46-s + 0.875·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 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
−7.50136799669702123180560377799, −6.87231635746065961134156986184, −6.07990432071003204598796847667, −5.15049393566020071335799991802, −4.84132098018455846662836521172, −3.78973024595814975369272522800, −2.84788762764959730109880934230, −2.32358969255822247087366120152, −0.816254076128081240605604169498, 0,
0.816254076128081240605604169498, 2.32358969255822247087366120152, 2.84788762764959730109880934230, 3.78973024595814975369272522800, 4.84132098018455846662836521172, 5.15049393566020071335799991802, 6.07990432071003204598796847667, 6.87231635746065961134156986184, 7.50136799669702123180560377799
