| L(s) = 1 | + i·2-s − 4-s − 2.34·5-s + (1.07 − 2.41i)7-s − i·8-s − 2.34i·10-s + 5.67i·11-s + 1.71i·13-s + (2.41 + 1.07i)14-s + 16-s − 1.76·17-s − 1.13i·19-s + 2.34·20-s − 5.67·22-s − 3.67i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.05·5-s + (0.407 − 0.913i)7-s − 0.353i·8-s − 0.742i·10-s + 1.71i·11-s + 0.477i·13-s + (0.645 + 0.288i)14-s + 0.250·16-s − 0.429·17-s − 0.261i·19-s + 0.525·20-s − 1.21·22-s − 0.766i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1833610342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1833610342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.07 + 2.41i)T \) |
| good | 5 | \( 1 + 2.34T + 5T^{2} \) |
| 11 | \( 1 - 5.67iT - 11T^{2} \) |
| 13 | \( 1 - 1.71iT - 13T^{2} \) |
| 17 | \( 1 + 1.76T + 17T^{2} \) |
| 19 | \( 1 + 1.13iT - 19T^{2} \) |
| 23 | \( 1 + 3.67iT - 23T^{2} \) |
| 29 | \( 1 + 4.15iT - 29T^{2} \) |
| 31 | \( 1 + 8.37iT - 31T^{2} \) |
| 37 | \( 1 + 9.19T + 37T^{2} \) |
| 41 | \( 1 + 7.99T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 2.22T + 59T^{2} \) |
| 61 | \( 1 + 8.99iT - 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 4.52iT - 71T^{2} \) |
| 73 | \( 1 - 5.34iT - 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 1.16T + 89T^{2} \) |
| 97 | \( 1 - 4.59iT - 97T^{2} \) |
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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
−9.597926506005121562688345989069, −8.411179635667932403102731256781, −7.83545225288596647498053894479, −7.05689384779793409809603157006, −6.59863595016117220387414706266, −4.99487470573521328268446860782, −4.42092708325414867963690729192, −3.72727268969986326024502682814, −1.93524994959243978490890612526, −0.080957174143444571344942062654,
1.55819210642587531063418747821, 3.14455909757288295787945443680, 3.52927943482471370142560463235, 4.93058112681097500120492916944, 5.58289089268559796335378909285, 6.75337874571808270377007664773, 8.017567436872313770368685920243, 8.473339148852923191237453574264, 9.063272852453572213779544086492, 10.31433574348684747514314077925
