| L(s) = 1 | + 3.09i·3-s − 0.561·5-s + (−1.40 − 2.24i)7-s − 6.59·9-s + 5.17i·11-s + 6.03i·13-s − 1.73i·15-s − 0.798·17-s + 3.89·19-s + (6.94 − 4.34i)21-s + (−1.78 − 4.45i)23-s − 4.68·25-s − 11.1i·27-s − 2.40·29-s − 4.50i·31-s + ⋯ |
| L(s) = 1 | + 1.78i·3-s − 0.250·5-s + (−0.530 − 0.847i)7-s − 2.19·9-s + 1.56i·11-s + 1.67i·13-s − 0.448i·15-s − 0.193·17-s + 0.894·19-s + (1.51 − 0.947i)21-s + (−0.371 − 0.928i)23-s − 0.937·25-s − 2.14i·27-s − 0.446·29-s − 0.809i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5567638917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5567638917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.40 + 2.24i)T \) |
| 23 | \( 1 + (1.78 + 4.45i)T \) |
| good | 3 | \( 1 - 3.09iT - 3T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 11 | \( 1 - 5.17iT - 11T^{2} \) |
| 13 | \( 1 - 6.03iT - 13T^{2} \) |
| 17 | \( 1 + 0.798T + 17T^{2} \) |
| 19 | \( 1 - 3.89T + 19T^{2} \) |
| 29 | \( 1 + 2.40T + 29T^{2} \) |
| 31 | \( 1 + 4.50iT - 31T^{2} \) |
| 37 | \( 1 + 10.6iT - 37T^{2} \) |
| 41 | \( 1 - 6.48iT - 41T^{2} \) |
| 43 | \( 1 + 3.09iT - 43T^{2} \) |
| 47 | \( 1 + 4.00iT - 47T^{2} \) |
| 53 | \( 1 - 6.55iT - 53T^{2} \) |
| 59 | \( 1 + 7.75iT - 59T^{2} \) |
| 61 | \( 1 + 9.53T + 61T^{2} \) |
| 67 | \( 1 - 4.44iT - 67T^{2} \) |
| 71 | \( 1 - 0.280T + 71T^{2} \) |
| 73 | \( 1 - 3.85iT - 73T^{2} \) |
| 79 | \( 1 + 10.0iT - 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 + 0.465T + 97T^{2} \) |
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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
−9.989325740254640662530591657846, −9.533012314105774893035268635832, −9.035861467215853384316738626513, −7.72727175412295116036262311535, −6.95046359527313428482357436166, −5.90484304309868828234188237061, −4.66528021609214790834236738984, −4.26351527451281226814648835944, −3.62240225796761002348972664700, −2.19987767589854459697557668528,
0.23053782455629289437213633276, 1.44208092917685372589080796748, 2.89495330919214222556572450632, 3.28479219729912179557806266105, 5.56091602934850016654488676099, 5.72309761162678476724160113782, 6.64627781755915846634667353719, 7.64100498623814371578268274829, 8.153984974547240963724293338877, 8.775689015926683447897748281986
