| L(s) = 1 | + 4·7-s − 2·11-s − 13-s + 4·17-s + 2·19-s + 2·23-s − 6·29-s − 8·31-s + 6·37-s − 10·41-s + 4·43-s + 9·49-s − 6·53-s + 6·59-s + 2·61-s + 4·67-s − 12·71-s − 2·73-s − 8·77-s − 8·79-s − 12·83-s − 14·89-s − 4·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.603·11-s − 0.277·13-s + 0.970·17-s + 0.458·19-s + 0.417·23-s − 1.11·29-s − 1.43·31-s + 0.986·37-s − 1.56·41-s + 0.609·43-s + 9/7·49-s − 0.824·53-s + 0.781·59-s + 0.256·61-s + 0.488·67-s − 1.42·71-s − 0.234·73-s − 0.911·77-s − 0.900·79-s − 1.31·83-s − 1.48·89-s − 0.419·91-s + 1.01·97-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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.65694720169628, −14.45408731877950, −14.03953913126136, −13.13846046495677, −12.96156445692564, −12.23696273575193, −11.63100766959262, −11.24019931315008, −10.90770209301804, −10.01677255676058, −9.878884284471497, −8.886046558413451, −8.623399264993786, −7.837176311097393, −7.476780408035060, −7.185809531370579, −6.105281322382366, −5.526863106452749, −5.140720283227583, −4.617614480652156, −3.846307732415879, −3.208279513858964, −2.410679753351037, −1.707810915156248, −1.138549914690416, 0,
1.138549914690416, 1.707810915156248, 2.410679753351037, 3.208279513858964, 3.846307732415879, 4.617614480652156, 5.140720283227583, 5.526863106452749, 6.105281322382366, 7.185809531370579, 7.476780408035060, 7.837176311097393, 8.623399264993786, 8.886046558413451, 9.878884284471497, 10.01677255676058, 10.90770209301804, 11.24019931315008, 11.63100766959262, 12.23696273575193, 12.96156445692564, 13.13846046495677, 14.03953913126136, 14.45408731877950, 14.65694720169628
