| L(s) = 1 | + 3-s − 3i·5-s + 9-s − 3i·15-s + 17-s − 4·23-s − 4·25-s + 27-s + 3·29-s − 8i·31-s − 5i·37-s − 3i·41-s − 4·43-s − 3i·45-s − 8i·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34i·5-s + 0.333·9-s − 0.774i·15-s + 0.242·17-s − 0.834·23-s − 0.800·25-s + 0.192·27-s + 0.557·29-s − 1.43i·31-s − 0.821i·37-s − 0.468i·41-s − 0.609·43-s − 0.447i·45-s − 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.879936958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.879936958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 + 5iT - 37T^{2} \) |
| 41 | \( 1 + 3iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 13T + 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 - 15T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8iT - 71T^{2} \) |
| 73 | \( 1 - 3iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.189184634703033602973399759141, −7.75162725156854576490049156413, −6.78290486579060523317340953452, −5.82707782060081975069638774171, −5.16105916834161185473654892066, −4.30795016481760430238987107373, −3.72239281218278114928859809148, −2.50964986626511634023770713047, −1.61168947646378694768543394246, −0.48705822446828856356673527654,
1.43426421156518844878516825165, 2.56538218718704606359029221767, 3.12603792166247840551936944068, 3.91059215706329356940367123656, 4.87866693708687105734821872127, 5.88622136568464825008302966386, 6.73872052276482494096933791082, 7.05666276638728578100557202365, 8.093429542561579956402647341956, 8.425129584193236544597750577012
