| L(s) = 1 | − 5-s − 4·11-s − 5·13-s − 5·17-s + 8·19-s − 4·23-s − 4·25-s + 3·29-s − 4·31-s + 3·37-s − 6·41-s + 4·43-s − 12·47-s − 7·49-s − 10·53-s + 4·55-s + 8·59-s − 5·61-s + 5·65-s + 8·67-s + 16·71-s − 5·73-s + 4·79-s + 4·83-s + 5·85-s + 3·89-s − 8·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 1.38·13-s − 1.21·17-s + 1.83·19-s − 0.834·23-s − 4/5·25-s + 0.557·29-s − 0.718·31-s + 0.493·37-s − 0.937·41-s + 0.609·43-s − 1.75·47-s − 49-s − 1.37·53-s + 0.539·55-s + 1.04·59-s − 0.640·61-s + 0.620·65-s + 0.977·67-s + 1.89·71-s − 0.585·73-s + 0.450·79-s + 0.439·83-s + 0.542·85-s + 0.317·89-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.971901975673958204405109485110, −9.485156034711632328618611283369, −8.106724387015710755402307071258, −7.65368189622279550088665370255, −6.66896824235989322509381509585, −5.36147470189881314610699768949, −4.67782322662683851894252888537, −3.32266987288345429770967410054, −2.19109980611617118550596768162, 0,
2.19109980611617118550596768162, 3.32266987288345429770967410054, 4.67782322662683851894252888537, 5.36147470189881314610699768949, 6.66896824235989322509381509585, 7.65368189622279550088665370255, 8.106724387015710755402307071258, 9.485156034711632328618611283369, 9.971901975673958204405109485110
