| L(s) = 1 | − 5-s − 4·11-s + 6·13-s − 2·17-s − 8·19-s + 4·23-s + 25-s − 6·29-s + 4·31-s − 2·37-s − 2·41-s + 12·43-s − 2·53-s + 4·55-s + 4·59-s − 6·61-s − 6·65-s + 4·67-s + 8·71-s − 6·73-s + 16·79-s + 4·83-s + 2·85-s − 18·89-s + 8·95-s − 6·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s + 1.66·13-s − 0.485·17-s − 1.83·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.312·41-s + 1.82·43-s − 0.274·53-s + 0.539·55-s + 0.520·59-s − 0.768·61-s − 0.744·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s + 1.80·79-s + 0.439·83-s + 0.216·85-s − 1.90·89-s + 0.820·95-s − 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−15.34170683712690, −14.85893394678361, −14.05860632783146, −13.55508234676565, −13.00778842718743, −12.77049380450403, −12.14085493012973, −11.18476872589306, −11.01440441214585, −10.66345372448543, −10.00012195118884, −9.064910258768872, −8.813368034108713, −8.129237830029414, −7.820379168727310, −6.961175841356922, −6.472718091780974, −5.861449443134204, −5.270323762616004, −4.474851973593114, −4.013253888200371, −3.334512061124414, −2.565755848358268, −1.912078017223615, −0.9077325511467445, 0,
0.9077325511467445, 1.912078017223615, 2.565755848358268, 3.334512061124414, 4.013253888200371, 4.474851973593114, 5.270323762616004, 5.861449443134204, 6.472718091780974, 6.961175841356922, 7.820379168727310, 8.129237830029414, 8.813368034108713, 9.064910258768872, 10.00012195118884, 10.66345372448543, 11.01440441214585, 11.18476872589306, 12.14085493012973, 12.77049380450403, 13.00778842718743, 13.55508234676565, 14.05860632783146, 14.85893394678361, 15.34170683712690
