| L(s) = 1 | + (1.75 + 1.38i)5-s + 0.408i·7-s − 5.98·11-s − 5.17i·13-s − 2.06i·17-s + 19-s + 4.32i·23-s + (1.16 + 4.86i)25-s + 8.83·29-s − 2.83·31-s + (−0.566 + 0.717i)35-s − 10.3i·37-s + 5.50·41-s − 8.61i·43-s − 1.25i·47-s + ⋯ |
| L(s) = 1 | + (0.785 + 0.619i)5-s + 0.154i·7-s − 1.80·11-s − 1.43i·13-s − 0.501i·17-s + 0.229·19-s + 0.901i·23-s + (0.232 + 0.972i)25-s + 1.64·29-s − 0.509·31-s + (−0.0956 + 0.121i)35-s − 1.69i·37-s + 0.858·41-s − 1.31i·43-s − 0.182i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.750050839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.750050839\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.75 - 1.38i)T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 0.408iT - 7T^{2} \) |
| 11 | \( 1 + 5.98T + 11T^{2} \) |
| 13 | \( 1 + 5.17iT - 13T^{2} \) |
| 17 | \( 1 + 2.06iT - 17T^{2} \) |
| 23 | \( 1 - 4.32iT - 23T^{2} \) |
| 29 | \( 1 - 8.83T + 29T^{2} \) |
| 31 | \( 1 + 2.83T + 31T^{2} \) |
| 37 | \( 1 + 10.3iT - 37T^{2} \) |
| 41 | \( 1 - 5.50T + 41T^{2} \) |
| 43 | \( 1 + 8.61iT - 43T^{2} \) |
| 47 | \( 1 + 1.25iT - 47T^{2} \) |
| 53 | \( 1 + 1.65iT - 53T^{2} \) |
| 59 | \( 1 - 5.02T + 59T^{2} \) |
| 61 | \( 1 + 2.14T + 61T^{2} \) |
| 67 | \( 1 + 5.02iT - 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 - 0.955T + 79T^{2} \) |
| 83 | \( 1 + 6.67iT - 83T^{2} \) |
| 89 | \( 1 + 3.81T + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.454480475464047047541739058966, −7.63639956321959748052059802041, −7.24353530171667505656941080090, −6.10664841852343051777079244984, −5.40864181866036078148217984444, −5.10481622631866821961175811785, −3.59084076862628067296105994013, −2.77227370035389245214037799858, −2.21652750694930749950950976423, −0.58957974566919118622576441844,
1.03817289928865530967769658143, 2.19309682977241132549001962749, 2.87273920584202898359894613672, 4.33134564091975923905779218384, 4.79570187138337033562747718281, 5.64252911176665884539203190962, 6.40291019511889095520050682021, 7.11834117390087308317739504384, 8.247829957946724367179528614334, 8.460819628591843147464497918990
