| L(s) = 1 | − 2-s + 4-s − 3.85·5-s − 3.23·7-s − 8-s + 3.85·10-s + 1.38·11-s − 2.85·13-s + 3.23·14-s + 16-s + 4.47·17-s + 4.47·19-s − 3.85·20-s − 1.38·22-s − 2.85·23-s + 9.85·25-s + 2.85·26-s − 3.23·28-s + 9.32·29-s + 7.38·31-s − 32-s − 4.47·34-s + 12.4·35-s − 37-s − 4.47·38-s + 3.85·40-s − 9.61·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.72·5-s − 1.22·7-s − 0.353·8-s + 1.21·10-s + 0.416·11-s − 0.791·13-s + 0.864·14-s + 0.250·16-s + 1.08·17-s + 1.02·19-s − 0.861·20-s − 0.294·22-s − 0.595·23-s + 1.97·25-s + 0.559·26-s − 0.611·28-s + 1.73·29-s + 1.32·31-s − 0.176·32-s − 0.766·34-s + 2.10·35-s − 0.164·37-s − 0.725·38-s + 0.609·40-s − 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5772984426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5772984426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 2.85T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 - 9.32T + 29T^{2} \) |
| 31 | \( 1 - 7.38T + 31T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 1.09T + 67T^{2} \) |
| 71 | \( 1 + 2.94T + 71T^{2} \) |
| 73 | \( 1 - 7.09T + 73T^{2} \) |
| 79 | \( 1 + 8.56T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 + 0.472T + 97T^{2} \) |
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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
−10.16084379052051402647932147893, −9.879949479711905537275271281386, −8.638109233435591597672701580597, −7.956053490775710793649126002140, −7.14044343528659801100496828380, −6.43945258485775047281224833849, −4.94521540255671770583730319194, −3.65445389378952927062629006311, −2.96807548835528318907132220424, −0.70293052423832835508495182842,
0.70293052423832835508495182842, 2.96807548835528318907132220424, 3.65445389378952927062629006311, 4.94521540255671770583730319194, 6.43945258485775047281224833849, 7.14044343528659801100496828380, 7.956053490775710793649126002140, 8.638109233435591597672701580597, 9.879949479711905537275271281386, 10.16084379052051402647932147893
