| L(s) = 1 | + 1.62·2-s + 0.653·4-s + 4.97·7-s − 2.19·8-s − 1.99·11-s − 0.308·13-s + 8.09·14-s − 4.87·16-s − 2.59·17-s − 19-s − 3.24·22-s + 7.88·23-s − 0.502·26-s + 3.24·28-s + 2.30·29-s + 9.53·31-s − 3.56·32-s − 4.22·34-s + 6.69·37-s − 1.62·38-s − 9.90·41-s + 8.52·43-s − 1.30·44-s + 12.8·46-s + 5.85·47-s + 17.7·49-s − 0.201·52-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.326·4-s + 1.87·7-s − 0.775·8-s − 0.601·11-s − 0.0855·13-s + 2.16·14-s − 1.21·16-s − 0.629·17-s − 0.229·19-s − 0.692·22-s + 1.64·23-s − 0.0984·26-s + 0.613·28-s + 0.428·29-s + 1.71·31-s − 0.629·32-s − 0.724·34-s + 1.10·37-s − 0.264·38-s − 1.54·41-s + 1.30·43-s − 0.196·44-s + 1.89·46-s + 0.853·47-s + 2.53·49-s − 0.0279·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.902722396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.902722396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.62T + 2T^{2} \) |
| 7 | \( 1 - 4.97T + 7T^{2} \) |
| 11 | \( 1 + 1.99T + 11T^{2} \) |
| 13 | \( 1 + 0.308T + 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 23 | \( 1 - 7.88T + 23T^{2} \) |
| 29 | \( 1 - 2.30T + 29T^{2} \) |
| 31 | \( 1 - 9.53T + 31T^{2} \) |
| 37 | \( 1 - 6.69T + 37T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 - 8.52T + 43T^{2} \) |
| 47 | \( 1 - 5.85T + 47T^{2} \) |
| 53 | \( 1 - 7.08T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 4.34T + 61T^{2} \) |
| 67 | \( 1 - 3.55T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 8.83T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 1.36T + 83T^{2} \) |
| 89 | \( 1 + 4.48T + 89T^{2} \) |
| 97 | \( 1 + 0.308T + 97T^{2} \) |
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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
−8.491721478257806445640165843508, −7.59541002649958560615808233749, −6.83938364719368976336968096135, −5.89764923612991914718466966356, −5.16752763411830171154582427902, −4.65229003873899104171950747818, −4.20589972724343042602525502704, −2.93239671181748526007549370604, −2.27784153741241130778730217485, −0.978997874799389923459429001693,
0.978997874799389923459429001693, 2.27784153741241130778730217485, 2.93239671181748526007549370604, 4.20589972724343042602525502704, 4.65229003873899104171950747818, 5.16752763411830171154582427902, 5.89764923612991914718466966356, 6.83938364719368976336968096135, 7.59541002649958560615808233749, 8.491721478257806445640165843508
