| L(s) = 1 | − 0.246·2-s − 0.801·3-s − 1.93·4-s + 0.198·6-s + 1.69·7-s + 0.972·8-s − 2.35·9-s − 0.911·11-s + 1.55·12-s + 1.55·13-s − 0.417·14-s + 3.63·16-s + 5.29·17-s + 0.582·18-s − 19-s − 1.35·21-s + 0.225·22-s + 4.24·23-s − 0.780·24-s − 0.384·26-s + 4.29·27-s − 3.28·28-s + 5.00·29-s + 1.82·31-s − 2.84·32-s + 0.731·33-s − 1.30·34-s + ⋯ |
| L(s) = 1 | − 0.174·2-s − 0.462·3-s − 0.969·4-s + 0.0808·6-s + 0.639·7-s + 0.343·8-s − 0.785·9-s − 0.274·11-s + 0.448·12-s + 0.431·13-s − 0.111·14-s + 0.909·16-s + 1.28·17-s + 0.137·18-s − 0.229·19-s − 0.296·21-s + 0.0480·22-s + 0.885·23-s − 0.159·24-s − 0.0753·26-s + 0.826·27-s − 0.620·28-s + 0.930·29-s + 0.328·31-s − 0.502·32-s + 0.127·33-s − 0.224·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9054111766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9054111766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.246T + 2T^{2} \) |
| 3 | \( 1 + 0.801T + 3T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 + 0.911T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 - 5.00T + 29T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 - 6.29T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 + 2.04T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 - 9.87T + 59T^{2} \) |
| 61 | \( 1 - 0.542T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 2.80T + 73T^{2} \) |
| 79 | \( 1 - 1.59T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 2.91T + 89T^{2} \) |
| 97 | \( 1 - 1.55T + 97T^{2} \) |
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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
−10.94964268241966714381145074329, −10.17807471108975932271771237286, −9.136005947361075249441790902075, −8.345102264079253792791639000254, −7.62669815188220381188037319731, −6.11045748435607278069387818145, −5.29469188982789780528686237166, −4.42912433186574816172859778924, −3.03624469790924194472196530940, −0.973117189691318859222470061420,
0.973117189691318859222470061420, 3.03624469790924194472196530940, 4.42912433186574816172859778924, 5.29469188982789780528686237166, 6.11045748435607278069387818145, 7.62669815188220381188037319731, 8.345102264079253792791639000254, 9.136005947361075249441790902075, 10.17807471108975932271771237286, 10.94964268241966714381145074329
