| L(s) = 1 | − 5-s − 4·11-s + 6·13-s − 2·17-s − 4·19-s − 8·23-s + 25-s − 6·29-s + 31-s − 2·37-s − 10·41-s + 4·43-s − 7·49-s + 10·53-s + 4·55-s − 12·59-s − 2·61-s − 6·65-s + 4·67-s + 2·73-s + 4·83-s + 2·85-s + 14·89-s + 4·95-s + 18·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.179·31-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 49-s + 1.37·53-s + 0.539·55-s − 1.56·59-s − 0.256·61-s − 0.744·65-s + 0.488·67-s + 0.234·73-s + 0.439·83-s + 0.216·85-s + 1.48·89-s + 0.410·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9363981897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9363981897\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.62105175662633, −15.12521115711406, −14.48574091859984, −13.71063713404797, −13.35540947810259, −12.96900070486287, −12.18294048446292, −11.75515165290939, −10.93109956469000, −10.73962012370373, −10.17061858055172, −9.362612050288573, −8.722227153780700, −8.140997062114677, −7.911190832059917, −7.033355918013313, −6.346626961796818, −5.875541536860035, −5.181683380800153, −4.384681735204807, −3.807019578842274, −3.236646545664651, −2.242323636413612, −1.646546967602663, −0.3786735759888804,
0.3786735759888804, 1.646546967602663, 2.242323636413612, 3.236646545664651, 3.807019578842274, 4.384681735204807, 5.181683380800153, 5.875541536860035, 6.346626961796818, 7.033355918013313, 7.911190832059917, 8.140997062114677, 8.722227153780700, 9.362612050288573, 10.17061858055172, 10.73962012370373, 10.93109956469000, 11.75515165290939, 12.18294048446292, 12.96900070486287, 13.35540947810259, 13.71063713404797, 14.48574091859984, 15.12521115711406, 15.62105175662633
