| L(s) = 1 | − i·2-s − 4-s + (−1.41 − 2.23i)7-s + i·8-s − 5.65i·11-s − 4.47i·13-s + (−2.23 + 1.41i)14-s + 16-s + 3.16·17-s − 3.16i·19-s − 5.65·22-s − 4i·23-s − 4.47·26-s + (1.41 + 2.23i)28-s + 2.82i·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.534 − 0.845i)7-s + 0.353i·8-s − 1.70i·11-s − 1.24i·13-s + (−0.597 + 0.377i)14-s + 0.250·16-s + 0.766·17-s − 0.725i·19-s − 1.20·22-s − 0.834i·23-s − 0.877·26-s + (0.267 + 0.422i)28-s + 0.525i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.180540751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.180540751\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.41 + 2.23i)T \) |
| good | 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 + 3.16iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 - 9.48iT - 61T^{2} \) |
| 67 | \( 1 - 7.07T + 67T^{2} \) |
| 71 | \( 1 + 1.41iT - 71T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.303098308938658329021185111857, −7.73408387207304427069643172755, −6.68871857480235029663548533388, −5.92022351094780232746267307405, −5.14330091496514857869600555765, −4.15390353081677441799385799665, −3.11754358940727775660071153124, −2.97491059785818326077798251071, −1.14640946304978255392084492053, −0.41344192160225552189843759391,
1.61269610049448858986037593532, 2.56401406705641685592371569511, 3.84727952996583726920311026144, 4.51117455396822307117557610711, 5.42967170470950844594149937766, 6.15473782074428545645702734383, 6.80979243944588292044876742203, 7.61891739103387380771587696128, 8.150903898339289162127817745653, 9.347790483446802875585042874379
