| L(s) = 1 | + 35.4·3-s + 405. i·5-s + 7.97e3i·7-s − 1.84e4·9-s − 2.03e4i·11-s + (−1.02e5 + 1.41e3i)13-s + 1.43e4i·15-s − 9.27e4·17-s − 5.47e5i·19-s + 2.82e5i·21-s + 1.27e6·23-s + 1.78e6·25-s − 1.34e6·27-s − 2.11e6·29-s − 3.12e6i·31-s + ⋯ |
| L(s) = 1 | + 0.252·3-s + 0.290i·5-s + 1.25i·7-s − 0.936·9-s − 0.419i·11-s + (−0.999 + 0.0136i)13-s + 0.0732i·15-s − 0.269·17-s − 0.963i·19-s + 0.316i·21-s + 0.948·23-s + 0.915·25-s − 0.488·27-s − 0.554·29-s − 0.607i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0136 + 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0136 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9367910752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9367910752\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (1.02e5 - 1.41e3i)T \) |
| good | 3 | \( 1 - 35.4T + 1.96e4T^{2} \) |
| 5 | \( 1 - 405. iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 7.97e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 + 2.03e4iT - 2.35e9T^{2} \) |
| 17 | \( 1 + 9.27e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.47e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 - 1.27e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.11e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.12e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 6.50e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 3.29e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 - 1.71e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.00e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 - 5.66e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.94e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 + 1.29e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 7.67e7iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 1.90e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + 2.21e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 2.74e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.37e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 8.62e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + 1.15e9iT - 7.60e17T^{2} \) |
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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
−11.70380513660621127212200719205, −10.82928414741686179230871877183, −9.247943577538218491780556420476, −8.713345955348975661869297456705, −7.31277023347088749629421735088, −5.97395055626514092423737074111, −4.95368769739616117586242633615, −3.05232825359183180074530347603, −2.30934012070915006252819273510, −0.25009286021006087665418645069,
1.12108293663255994012948618168, 2.73331948644735225317770583808, 4.10460745481819157371286571172, 5.29621361063004665002264375279, 6.87210028455903579571710705602, 7.81994010535296606531092208784, 9.019150014430056858902026535133, 10.13322515943390791873913312642, 11.11218959895678089749681204057, 12.32811812870908456637150208736
