| L(s) = 1 | − 30.0i·3-s + 25i·5-s + 45.4·7-s − 658.·9-s − 142. i·11-s − 897. i·13-s + 750.·15-s + 11.4·17-s + 1.81e3i·19-s − 1.36e3i·21-s − 3.87e3·23-s − 625·25-s + 1.24e4i·27-s + 3.46e3i·29-s − 1.55e3·31-s + ⋯ |
| L(s) = 1 | − 1.92i·3-s + 0.447i·5-s + 0.350·7-s − 2.70·9-s − 0.355i·11-s − 1.47i·13-s + 0.861·15-s + 0.00964·17-s + 1.15i·19-s − 0.675i·21-s − 1.52·23-s − 0.200·25-s + 3.29i·27-s + 0.764i·29-s − 0.289·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.01293959086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01293959086\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25iT \) |
| good | 3 | \( 1 + 30.0iT - 243T^{2} \) |
| 7 | \( 1 - 45.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 142. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 897. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 11.4T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.81e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.87e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.46e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.61e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.48e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.32e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.90e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.25e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 652. iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 4.71e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 4.63e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.53e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.24e4T + 8.58e9T^{2} \) |
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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
−11.17244959286391372524899336482, −10.29157849599949515085088365548, −8.659973672268637232543389486481, −7.891493289273937647390027820452, −7.34771865180356783934703693605, −6.11863537985661730747237568595, −5.58858850171765886294077236087, −3.42427010304379260191520080691, −2.29574393430344780455877861510, −1.21640516625785591145220218716,
0.00341728204513170318580187366, 2.19128296732849287556235814977, 3.74144792584067199636818288793, 4.50594969953705116820627788118, 5.20735171618854972500427423300, 6.48624158207924985536059089194, 8.138741969070805758060256893351, 9.020383495485476660524722922567, 9.636393514623307363853619384047, 10.44790606515421502116062210230
