| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 4·11-s + 2·13-s − 14-s + 16-s + 2·17-s − 4·19-s − 4·22-s − 8·23-s + 2·26-s − 28-s − 6·29-s − 8·31-s + 32-s + 2·34-s + 2·37-s − 4·38-s − 2·41-s + 12·43-s − 4·44-s − 8·46-s − 8·47-s + 49-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.20·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.852·22-s − 1.66·23-s + 0.392·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.328·37-s − 0.648·38-s − 0.312·41-s + 1.82·43-s − 0.603·44-s − 1.17·46-s − 1.16·47-s + 1/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.038242046145279632455890814962, −7.64148986041460665692018995996, −6.65484672323719867937445615446, −5.82845148448746450571710928831, −5.42910934483244127499298506442, −4.27349277849080285060566701899, −3.66224755981270835231661103930, −2.66136363605390835360055298001, −1.78234535695426588996284856767, 0,
1.78234535695426588996284856767, 2.66136363605390835360055298001, 3.66224755981270835231661103930, 4.27349277849080285060566701899, 5.42910934483244127499298506442, 5.82845148448746450571710928831, 6.65484672323719867937445615446, 7.64148986041460665692018995996, 8.038242046145279632455890814962
