| L(s) = 1 | + 37·7-s − 89·13-s − 163·19-s − 289·31-s − 110·37-s − 71·43-s + 1.02e3·49-s + 719·61-s − 1.00e3·67-s − 1.19e3·73-s + 884·79-s − 3.29e3·91-s + 523·97-s − 1.82e3·103-s − 1.56e3·109-s + ⋯ |
| L(s) = 1 | + 1.99·7-s − 1.89·13-s − 1.96·19-s − 1.67·31-s − 0.488·37-s − 0.251·43-s + 2.99·49-s + 1.50·61-s − 1.83·67-s − 1.90·73-s + 1.25·79-s − 3.79·91-s + 0.547·97-s − 1.74·103-s − 1.37·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 37 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 89 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 163 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 289 T + p^{3} T^{2} \) |
| 37 | \( 1 + 110 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 + 71 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 719 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1007 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 1190 T + p^{3} T^{2} \) |
| 79 | \( 1 - 884 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 - 523 T + p^{3} T^{2} \) |
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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
−9.171533862382586125532885984426, −8.403312877624171770795674139021, −7.65328836703601654242388743001, −6.95169254357698925660683999838, −5.56270409500053343461452039716, −4.83098389905645347601886373734, −4.14136905628284483714785641355, −2.40531407293645915885499211955, −1.70429886250894445904463500386, 0,
1.70429886250894445904463500386, 2.40531407293645915885499211955, 4.14136905628284483714785641355, 4.83098389905645347601886373734, 5.56270409500053343461452039716, 6.95169254357698925660683999838, 7.65328836703601654242388743001, 8.403312877624171770795674139021, 9.171533862382586125532885984426
