| L(s) = 1 | − i·2-s − 4-s + 0.575i·5-s + (−2.19 + 1.47i)7-s + i·8-s + 0.575·10-s + 5.11·11-s − 6.37·13-s + (1.47 + 2.19i)14-s + 16-s − 1.63i·17-s + (1.26 − 4.17i)19-s − 0.575i·20-s − 5.11i·22-s − 6.12·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.257i·5-s + (−0.831 + 0.556i)7-s + 0.353i·8-s + 0.182·10-s + 1.54·11-s − 1.76·13-s + (0.393 + 0.587i)14-s + 0.250·16-s − 0.395i·17-s + (0.290 − 0.956i)19-s − 0.128i·20-s − 1.08i·22-s − 1.27·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6010778248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6010778248\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.19 - 1.47i)T \) |
| 19 | \( 1 + (-1.26 + 4.17i)T \) |
| good | 5 | \( 1 - 0.575iT - 5T^{2} \) |
| 11 | \( 1 - 5.11T + 11T^{2} \) |
| 13 | \( 1 + 6.37T + 13T^{2} \) |
| 17 | \( 1 + 1.63iT - 17T^{2} \) |
| 23 | \( 1 + 6.12T + 23T^{2} \) |
| 29 | \( 1 - 4.39iT - 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 - 7.51iT - 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 + 4.73iT - 47T^{2} \) |
| 53 | \( 1 + 11.7iT - 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 14.8iT - 61T^{2} \) |
| 67 | \( 1 + 7.14iT - 67T^{2} \) |
| 71 | \( 1 + 11.7iT - 71T^{2} \) |
| 73 | \( 1 + 6.47iT - 73T^{2} \) |
| 79 | \( 1 + 7.82iT - 79T^{2} \) |
| 83 | \( 1 - 11.3iT - 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 8.35T + 97T^{2} \) |
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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
−8.920982161746243508496540017384, −7.972566834984729666574058089297, −6.75469791828653611229083963260, −6.60355422632485034102988569335, −5.22834587341416027243137227130, −4.60549415253930505208187429667, −3.44384689273485635467531669139, −2.80963282539986709691829596869, −1.78606850784568661191373488599, −0.21305840903470937164560388646,
1.25758550000909736403796704813, 2.75342465755501713977327296093, 4.02711202717619838222145821765, 4.34107599597572763064344538081, 5.62333506194174146179374664107, 6.25094023394849737549037572202, 7.05552295920925033811526339202, 7.57819705440061927025879662133, 8.519360712251226013376807060347, 9.346520623884840822985697630337
