| L(s) = 1 | + (−1.08 − 1.95i)5-s + (−0.595 + 2.57i)7-s − 3.74i·11-s + 3.36·13-s − 0.841i·17-s + 5.59i·19-s − 2.35·23-s + (−2.64 + 4.24i)25-s − 1.41i·29-s + 8.66i·31-s + (5.68 − 1.63i)35-s − 5.15i·37-s − 5.74·41-s − 3.32i·43-s + 6.43i·47-s + ⋯ |
| L(s) = 1 | + (−0.485 − 0.874i)5-s + (−0.224 + 0.974i)7-s − 1.12i·11-s + 0.931·13-s − 0.204i·17-s + 1.28i·19-s − 0.490·23-s + (−0.529 + 0.848i)25-s − 0.262i·29-s + 1.55i·31-s + (0.961 − 0.276i)35-s − 0.847i·37-s − 0.896·41-s − 0.507i·43-s + 0.938i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.153921285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.153921285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.08 + 1.95i)T \) |
| 7 | \( 1 + (0.595 - 2.57i)T \) |
| good | 11 | \( 1 + 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 0.841iT - 17T^{2} \) |
| 19 | \( 1 - 5.59iT - 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 8.66iT - 31T^{2} \) |
| 37 | \( 1 + 5.15iT - 37T^{2} \) |
| 41 | \( 1 + 5.74T + 41T^{2} \) |
| 43 | \( 1 + 3.32iT - 43T^{2} \) |
| 47 | \( 1 - 6.43iT - 47T^{2} \) |
| 53 | \( 1 + 9.64T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 1.82iT - 67T^{2} \) |
| 71 | \( 1 + 3.74iT - 71T^{2} \) |
| 73 | \( 1 + 0.979T + 73T^{2} \) |
| 79 | \( 1 - 6.58T + 79T^{2} \) |
| 83 | \( 1 - 12.5iT - 83T^{2} \) |
| 89 | \( 1 - 2.16T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.293118074447628300513162997027, −8.071932448132345810828866439062, −6.87654195117522990356744455255, −5.96819606320077133455496377796, −5.62141713318959656676001572690, −4.78135126457935533192947180955, −3.70234307773047837654972325859, −3.29307543627120156049389651873, −1.98182255280469411920741699538, −0.988103857673195488527644118998,
0.36085388873079958469933448818, 1.73080672900616336505159812420, 2.77045402467645248325732717635, 3.66261900760826416067131062159, 4.22567468236714643564685629827, 5.00823042720014422877238409471, 6.25662334169646979610153047801, 6.68180627639594018885298781814, 7.35504984786174474044869700848, 7.905908134366405408970639543752
