| L(s) = 1 | + 0.890·5-s + 1.91i·7-s + 0.349i·11-s + 2.26i·13-s + 5.65i·17-s + 3.17i·19-s − 7.20·23-s − 4.20·25-s − 7.92i·29-s − 7.20·31-s + 1.70i·35-s − 0.890·37-s + (−5.98 − 2.26i)41-s + 3.20·43-s − 7.96i·47-s + ⋯ |
| L(s) = 1 | + 0.398·5-s + 0.725i·7-s + 0.105i·11-s + 0.629i·13-s + 1.37i·17-s + 0.729i·19-s − 1.50·23-s − 0.841·25-s − 1.47i·29-s − 1.29·31-s + 0.288i·35-s − 0.146·37-s + (−0.935 − 0.354i)41-s + 0.489·43-s − 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7854634316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7854634316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 + (5.98 + 2.26i)T \) |
| good | 5 | \( 1 - 0.890T + 5T^{2} \) |
| 7 | \( 1 - 1.91iT - 7T^{2} \) |
| 11 | \( 1 - 0.349iT - 11T^{2} \) |
| 13 | \( 1 - 2.26iT - 13T^{2} \) |
| 17 | \( 1 - 5.65iT - 17T^{2} \) |
| 19 | \( 1 - 3.17iT - 19T^{2} \) |
| 23 | \( 1 + 7.20T + 23T^{2} \) |
| 29 | \( 1 + 7.92iT - 29T^{2} \) |
| 31 | \( 1 + 7.20T + 31T^{2} \) |
| 37 | \( 1 + 0.890T + 37T^{2} \) |
| 43 | \( 1 - 3.20T + 43T^{2} \) |
| 47 | \( 1 + 7.96iT - 47T^{2} \) |
| 53 | \( 1 + 0.871iT - 53T^{2} \) |
| 59 | \( 1 + 6.76T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 4.18iT - 67T^{2} \) |
| 71 | \( 1 + 6.56iT - 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 7.96iT - 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 15.2iT - 89T^{2} \) |
| 97 | \( 1 + 17.2iT - 97T^{2} \) |
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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
−9.018628342356152591065567183130, −8.360230863742309589234952538232, −7.71473183770833846216911557445, −6.68726931679487677071336597764, −5.84568090400824501686398782384, −5.59305949138896811668480204836, −4.20180086528257394120262454379, −3.70304200055388096047382611849, −2.21475311442167299123617915516, −1.78425118558934665123218212010,
0.22800433952503795983001679571, 1.54204429953375348672961925172, 2.69657860796741669079333152673, 3.58417996084018185879304346484, 4.52973174924530753480531140915, 5.36663213707048246970340656735, 6.05880272194766214941004324325, 7.12293951829014368176869008656, 7.46146973363950765501854814551, 8.438463012465333054676204362456
