L(s) = 1 | + (1 − 2i)5-s + 4i·7-s − 4i·13-s − 8·19-s − 4i·23-s + (−3 − 4i)25-s + 6·29-s − 8·31-s + (8 + 4i)35-s − 4i·37-s − 6·41-s − 4i·43-s − 4i·47-s − 9·49-s − 12i·53-s + ⋯ |
L(s) = 1 | + (0.447 − 0.894i)5-s + 1.51i·7-s − 1.10i·13-s − 1.83·19-s − 0.834i·23-s + (−0.600 − 0.800i)25-s + 1.11·29-s − 1.43·31-s + (1.35 + 0.676i)35-s − 0.657i·37-s − 0.937·41-s − 0.609i·43-s − 0.583i·47-s − 1.28·49-s − 1.64i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.101060931\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.101060931\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 + 2i)T \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.485713043202305434625108762408, −8.248024791393879519043058905453, −6.86228328144141468630721305600, −6.06695411497513431143876353255, −5.43321844741185972155164830036, −4.86338283024674805455312110167, −3.76332144322671275734759603084, −2.51823609194052231496229452968, −1.93151085149406377507753616510, −0.33245930307245091301020860577,
1.39693010110303161887956545633, 2.36328291166322291582718703481, 3.57551077336006028217559565350, 4.14268417525768013317137248668, 5.06118418808619596009720933550, 6.39305521021963872204117553144, 6.60655887452073018710190649833, 7.39313239260340501655271341472, 8.122259736369629121132686590431, 9.167855520859607630918541059991