L(s) = 1 | + 2·7-s + 2·11-s + 13-s + 2·17-s + 4·19-s − 4·29-s − 8·31-s − 6·37-s + 6·41-s + 4·43-s + 8·47-s − 3·49-s + 2·53-s + 10·59-s − 14·61-s − 16·67-s − 4·71-s − 8·73-s + 4·77-s + 8·79-s − 12·83-s − 6·89-s + 2·91-s − 12·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 0.603·11-s + 0.277·13-s + 0.485·17-s + 0.917·19-s − 0.742·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s + 1.30·59-s − 1.79·61-s − 1.95·67-s − 0.474·71-s − 0.936·73-s + 0.455·77-s + 0.900·79-s − 1.31·83-s − 0.635·89-s + 0.209·91-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.86296620205901, −14.30804986017296, −13.92351911421268, −13.42551418613332, −12.71099027344217, −12.27360736301868, −11.70982890369885, −11.29395830212033, −10.74179244012259, −10.28912641244066, −9.483419452647985, −9.113370031000946, −8.645972103064215, −7.892000273599716, −7.378324317348456, −7.099275568739830, −6.124379453457288, −5.653160559720422, −5.205877613117178, −4.392234887699935, −3.886563764731248, −3.254203129565030, −2.485452515125637, −1.564370135697444, −1.210337576569193, 0,
1.210337576569193, 1.564370135697444, 2.485452515125637, 3.254203129565030, 3.886563764731248, 4.392234887699935, 5.205877613117178, 5.653160559720422, 6.124379453457288, 7.099275568739830, 7.378324317348456, 7.892000273599716, 8.645972103064215, 9.113370031000946, 9.483419452647985, 10.28912641244066, 10.74179244012259, 11.29395830212033, 11.70982890369885, 12.27360736301868, 12.71099027344217, 13.42551418613332, 13.92351911421268, 14.30804986017296, 14.86296620205901