| L(s) = 1 | − 4·5-s + 13-s − 3·17-s − 6·19-s + 7·23-s + 11·25-s − 9·29-s + 8·31-s + 2·37-s + 8·41-s + 7·43-s + 2·47-s − 7·49-s + 9·53-s + 4·59-s + 7·61-s − 4·65-s − 8·67-s − 14·71-s + 14·73-s + 5·79-s − 2·83-s + 12·85-s + 24·95-s − 4·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.277·13-s − 0.727·17-s − 1.37·19-s + 1.45·23-s + 11/5·25-s − 1.67·29-s + 1.43·31-s + 0.328·37-s + 1.24·41-s + 1.06·43-s + 0.291·47-s − 49-s + 1.23·53-s + 0.520·59-s + 0.896·61-s − 0.496·65-s − 0.977·67-s − 1.66·71-s + 1.63·73-s + 0.562·79-s − 0.219·83-s + 1.30·85-s + 2.46·95-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.08560551291501, −12.63655605428560, −12.34378622474458, −11.62872864165087, −11.26125944260169, −11.06118924484871, −10.58268413820921, −10.01068428406880, −9.095342933972631, −9.018082399431156, −8.423410613631972, −7.991034755490971, −7.512398903424604, −7.069806775527618, −6.626899416015057, −6.063632297236967, −5.417608524561375, −4.656706438831828, −4.417568563494208, −3.873558724142894, −3.496861810644455, −2.666437590578066, −2.354920643788847, −1.278321620359692, −0.6728150209483487, 0,
0.6728150209483487, 1.278321620359692, 2.354920643788847, 2.666437590578066, 3.496861810644455, 3.873558724142894, 4.417568563494208, 4.656706438831828, 5.417608524561375, 6.063632297236967, 6.626899416015057, 7.069806775527618, 7.512398903424604, 7.991034755490971, 8.423410613631972, 9.018082399431156, 9.095342933972631, 10.01068428406880, 10.58268413820921, 11.06118924484871, 11.26125944260169, 11.62872864165087, 12.34378622474458, 12.63655605428560, 13.08560551291501
