| L(s) = 1 | + 5-s + 13-s − 8·17-s + 2·19-s + 6·23-s + 25-s + 2·29-s + 4·31-s + 2·37-s − 8·41-s + 8·43-s − 8·47-s − 14·59-s − 2·61-s + 65-s + 12·67-s − 2·71-s − 2·73-s + 8·79-s + 16·83-s − 8·85-s + 2·95-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.277·13-s − 1.94·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.328·37-s − 1.24·41-s + 1.21·43-s − 1.16·47-s − 1.82·59-s − 0.256·61-s + 0.124·65-s + 1.46·67-s − 0.237·71-s − 0.234·73-s + 0.900·79-s + 1.75·83-s − 0.867·85-s + 0.205·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.10714500030233, −12.96337295137271, −12.17370259364245, −11.78607182791590, −11.21722297033579, −10.84529475575649, −10.50318609342710, −9.862446560937266, −9.300412186666133, −9.038621137534871, −8.584721903015287, −7.968021068247002, −7.522479504405805, −6.799797297623893, −6.455661607400590, −6.223993804150240, −5.297356072381384, −4.966342421553024, −4.512893337864477, −3.877534183304172, −3.218076222342991, −2.680505750501968, −2.166387353126396, −1.480801269759272, −0.8507191409501801, 0,
0.8507191409501801, 1.480801269759272, 2.166387353126396, 2.680505750501968, 3.218076222342991, 3.877534183304172, 4.512893337864477, 4.966342421553024, 5.297356072381384, 6.223993804150240, 6.455661607400590, 6.799797297623893, 7.522479504405805, 7.968021068247002, 8.584721903015287, 9.038621137534871, 9.300412186666133, 9.862446560937266, 10.50318609342710, 10.84529475575649, 11.21722297033579, 11.78607182791590, 12.17370259364245, 12.96337295137271, 13.10714500030233
