L(s) = 1 | − 3-s − 2·7-s + 9-s + 2·11-s + 2·13-s − 6·17-s + 8·19-s + 2·21-s − 4·23-s − 27-s − 8·29-s − 2·33-s − 10·37-s − 2·39-s + 2·41-s + 12·43-s − 3·49-s + 6·51-s + 10·53-s − 8·57-s − 6·59-s − 2·61-s − 2·63-s + 8·67-s + 4·69-s + 4·71-s − 4·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s + 0.436·21-s − 0.834·23-s − 0.192·27-s − 1.48·29-s − 0.348·33-s − 1.64·37-s − 0.320·39-s + 0.312·41-s + 1.82·43-s − 3/7·49-s + 0.840·51-s + 1.37·53-s − 1.05·57-s − 0.781·59-s − 0.256·61-s − 0.251·63-s + 0.977·67-s + 0.481·69-s + 0.474·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.74394170188589551682783177310, −7.10268607733195602713107348372, −6.44233769347313947127392684797, −5.79737460693636511532351524277, −5.10528381879689958194764029794, −4.01900235692065045959130332284, −3.55154087393956388493946443484, −2.35772885109969110936902696506, −1.25528436695132346499729645328, 0,
1.25528436695132346499729645328, 2.35772885109969110936902696506, 3.55154087393956388493946443484, 4.01900235692065045959130332284, 5.10528381879689958194764029794, 5.79737460693636511532351524277, 6.44233769347313947127392684797, 7.10268607733195602713107348372, 7.74394170188589551682783177310