| L(s) = 1 | − 2.45·2-s − 2.63·3-s + 4.02·4-s + 6.47·6-s − 2.50·7-s − 4.96·8-s + 3.96·9-s − 4.88·11-s − 10.6·12-s − 0.0357·13-s + 6.14·14-s + 4.14·16-s + 0.298·17-s − 9.72·18-s − 5.33·19-s + 6.60·21-s + 12.0·22-s + 6.84·23-s + 13.1·24-s + 0.0876·26-s − 2.54·27-s − 10.0·28-s − 5.22·29-s + 9.08·31-s − 0.238·32-s + 12.9·33-s − 0.733·34-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 1.52·3-s + 2.01·4-s + 2.64·6-s − 0.946·7-s − 1.75·8-s + 1.32·9-s − 1.47·11-s − 3.06·12-s − 0.00990·13-s + 1.64·14-s + 1.03·16-s + 0.0724·17-s − 2.29·18-s − 1.22·19-s + 1.44·21-s + 2.55·22-s + 1.42·23-s + 2.67·24-s + 0.0171·26-s − 0.488·27-s − 1.90·28-s − 0.970·29-s + 1.63·31-s − 0.0422·32-s + 2.24·33-s − 0.125·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01352058188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01352058188\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 3 | \( 1 + 2.63T + 3T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 + 4.88T + 11T^{2} \) |
| 13 | \( 1 + 0.0357T + 13T^{2} \) |
| 17 | \( 1 - 0.298T + 17T^{2} \) |
| 19 | \( 1 + 5.33T + 19T^{2} \) |
| 23 | \( 1 - 6.84T + 23T^{2} \) |
| 29 | \( 1 + 5.22T + 29T^{2} \) |
| 31 | \( 1 - 9.08T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 + 0.718T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 - 1.44T + 61T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 - 3.37T + 71T^{2} \) |
| 73 | \( 1 + 7.51T + 73T^{2} \) |
| 79 | \( 1 - 3.77T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−8.207537061592638921540086136888, −7.31256568830676443809301999406, −6.62832649539984063015032839925, −6.41312295629048931461090590334, −5.33329550370118954828399183797, −4.84578909495831904925899992607, −3.36469760559167894098861913691, −2.44026946698441227438442020689, −1.34205766175545602408840590970, −0.090585463654434576978768841374,
0.090585463654434576978768841374, 1.34205766175545602408840590970, 2.44026946698441227438442020689, 3.36469760559167894098861913691, 4.84578909495831904925899992607, 5.33329550370118954828399183797, 6.41312295629048931461090590334, 6.62832649539984063015032839925, 7.31256568830676443809301999406, 8.207537061592638921540086136888
