| L(s) = 1 | + 1.41·2-s − 1.99·4-s + 9i·7-s − 8.48·8-s + 18.3i·11-s + i·13-s + 12.7i·14-s − 4.00·16-s + 26.8·17-s − 25·19-s + 26i·22-s + 4.24·23-s + 1.41i·26-s − 17.9i·28-s + 26.8i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.499·4-s + 1.28i·7-s − 1.06·8-s + 1.67i·11-s + 0.0769i·13-s + 0.909i·14-s − 0.250·16-s + 1.58·17-s − 1.31·19-s + 1.18i·22-s + 0.184·23-s + 0.0543i·26-s − 0.642i·28-s + 0.926i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.960352 + 1.11847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.960352 + 1.11847i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.41T + 4T^{2} \) |
| 7 | \( 1 - 9iT - 49T^{2} \) |
| 11 | \( 1 - 18.3iT - 121T^{2} \) |
| 13 | \( 1 - iT - 169T^{2} \) |
| 17 | \( 1 - 26.8T + 289T^{2} \) |
| 19 | \( 1 + 25T + 361T^{2} \) |
| 23 | \( 1 - 4.24T + 529T^{2} \) |
| 29 | \( 1 - 26.8iT - 841T^{2} \) |
| 31 | \( 1 + 39T + 961T^{2} \) |
| 37 | \( 1 + 32iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 5.65iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 32.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 96.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 9.89iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 73T + 3.72e3T^{2} \) |
| 67 | \( 1 + 63iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 62.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 136iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 24T + 6.24e3T^{2} \) |
| 83 | \( 1 - 46.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 7iT - 9.40e3T^{2} \) |
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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
−12.65332991409113008401390650395, −11.74572996003332193039895952265, −10.22947812436824859330687022172, −9.315750499762115073302712231206, −8.504353338432426933027634306887, −7.12274354305100634448279968914, −5.74713748635186946928795010304, −5.01628279795112691035567279412, −3.75909257093194343656265699079, −2.22808630141681842520291173513,
0.66203721288307329801288480067, 3.29010064785278608502816317874, 4.08260669534730046180297217563, 5.41126079327799684409229033925, 6.36870648191100473249354892716, 7.81161061384981092440207150973, 8.701898940307724008625165957281, 9.949590094232477024298627650169, 10.83342756259383364404040163953, 11.86735927459109730195529965279
