| L(s) = 1 | − 0.141·2-s − 1.97·4-s + 0.858·7-s + 0.563·8-s + 3.67·11-s − 4.58·13-s − 0.121·14-s + 3.87·16-s − 5.30·17-s + 6.36·19-s − 0.521·22-s + 3.42·23-s + 0.649·26-s − 1.69·28-s − 3.73·29-s + 1.25·31-s − 1.67·32-s + 0.751·34-s + 7.45·37-s − 0.902·38-s + 2.53·41-s + 3.37·43-s − 7.28·44-s − 0.485·46-s − 8.49·47-s − 6.26·49-s + 9.07·52-s + ⋯ |
| L(s) = 1 | − 0.100·2-s − 0.989·4-s + 0.324·7-s + 0.199·8-s + 1.10·11-s − 1.27·13-s − 0.0325·14-s + 0.969·16-s − 1.28·17-s + 1.46·19-s − 0.111·22-s + 0.713·23-s + 0.127·26-s − 0.321·28-s − 0.693·29-s + 0.225·31-s − 0.296·32-s + 0.128·34-s + 1.22·37-s − 0.146·38-s + 0.395·41-s + 0.515·43-s − 1.09·44-s − 0.0715·46-s − 1.23·47-s − 0.894·49-s + 1.25·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.372152410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.372152410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.141T + 2T^{2} \) |
| 7 | \( 1 - 0.858T + 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 13 | \( 1 + 4.58T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 - 6.36T + 19T^{2} \) |
| 23 | \( 1 - 3.42T + 23T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 - 1.25T + 31T^{2} \) |
| 37 | \( 1 - 7.45T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 3.37T + 43T^{2} \) |
| 47 | \( 1 + 8.49T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 3.34T + 67T^{2} \) |
| 71 | \( 1 - 4.32T + 71T^{2} \) |
| 73 | \( 1 - 9.08T + 73T^{2} \) |
| 79 | \( 1 + 3.24T + 79T^{2} \) |
| 83 | \( 1 + 7.39T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.113974977747442220761779074314, −7.53047885887065381437376887208, −6.79964591368049178465886331889, −5.94806538509759279281473911841, −4.96779447787503217147585830515, −4.65411815114911878951727954214, −3.78955206961991850990536937232, −2.89012647474747905461983074841, −1.72035845958284085184918126020, −0.65815820433299664980996561691,
0.65815820433299664980996561691, 1.72035845958284085184918126020, 2.89012647474747905461983074841, 3.78955206961991850990536937232, 4.65411815114911878951727954214, 4.96779447787503217147585830515, 5.94806538509759279281473911841, 6.79964591368049178465886331889, 7.53047885887065381437376887208, 8.113974977747442220761779074314
