| L(s) = 1 | − 5-s − 2·7-s − 11-s + 3·17-s − 5·19-s + 6·23-s + 25-s + 6·29-s − 2·31-s + 2·35-s − 5·37-s + 9·41-s − 43-s − 3·47-s − 3·49-s + 55-s − 12·59-s + 8·61-s − 5·67-s + 12·71-s − 2·73-s + 2·77-s − 10·79-s − 6·83-s − 3·85-s + 5·95-s + 97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.301·11-s + 0.727·17-s − 1.14·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.359·31-s + 0.338·35-s − 0.821·37-s + 1.40·41-s − 0.152·43-s − 0.437·47-s − 3/7·49-s + 0.134·55-s − 1.56·59-s + 1.02·61-s − 0.610·67-s + 1.42·71-s − 0.234·73-s + 0.227·77-s − 1.12·79-s − 0.658·83-s − 0.325·85-s + 0.512·95-s + 0.101·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.76996380961571, −12.50535445677601, −11.96489640177151, −11.44630118672657, −10.91944291758720, −10.59554194429889, −10.15327840808273, −9.571055642949849, −9.240812533407727, −8.574650182682074, −8.344047758606645, −7.724208924908072, −7.242258149304066, −6.772588520776951, −6.383625073660858, −5.807903485219996, −5.325730079393561, −4.657943928030600, −4.367633662187514, −3.619068782567359, −3.175174363476593, −2.783148848356604, −2.101819061773361, −1.342010656346573, −0.6784122740182651, 0,
0.6784122740182651, 1.342010656346573, 2.101819061773361, 2.783148848356604, 3.175174363476593, 3.619068782567359, 4.367633662187514, 4.657943928030600, 5.325730079393561, 5.807903485219996, 6.383625073660858, 6.772588520776951, 7.242258149304066, 7.724208924908072, 8.344047758606645, 8.574650182682074, 9.240812533407727, 9.571055642949849, 10.15327840808273, 10.59554194429889, 10.91944291758720, 11.44630118672657, 11.96489640177151, 12.50535445677601, 12.76996380961571
