| L(s) = 1 | + 2.18·2-s − 3-s + 2.78·4-s − 5-s − 2.18·6-s + 2.50·7-s + 1.71·8-s + 9-s − 2.18·10-s + 2.90·11-s − 2.78·12-s + 4.71·13-s + 5.47·14-s + 15-s − 1.81·16-s − 3.69·17-s + 2.18·18-s + 2.35·19-s − 2.78·20-s − 2.50·21-s + 6.35·22-s + 2.09·23-s − 1.71·24-s + 25-s + 10.3·26-s − 27-s + 6.96·28-s + ⋯ |
| L(s) = 1 | + 1.54·2-s − 0.577·3-s + 1.39·4-s − 0.447·5-s − 0.893·6-s + 0.945·7-s + 0.607·8-s + 0.333·9-s − 0.691·10-s + 0.875·11-s − 0.804·12-s + 1.30·13-s + 1.46·14-s + 0.258·15-s − 0.452·16-s − 0.895·17-s + 0.515·18-s + 0.541·19-s − 0.622·20-s − 0.545·21-s + 1.35·22-s + 0.436·23-s − 0.350·24-s + 0.200·25-s + 2.02·26-s − 0.192·27-s + 1.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.680971482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.680971482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 13 | \( 1 - 4.71T + 13T^{2} \) |
| 17 | \( 1 + 3.69T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 - 2.09T + 23T^{2} \) |
| 29 | \( 1 - 7.85T + 29T^{2} \) |
| 31 | \( 1 + 7.35T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 4.79T + 47T^{2} \) |
| 53 | \( 1 - 2.28T + 53T^{2} \) |
| 59 | \( 1 - 3.66T + 59T^{2} \) |
| 61 | \( 1 - 8.96T + 61T^{2} \) |
| 67 | \( 1 - 7.96T + 67T^{2} \) |
| 71 | \( 1 + 8.68T + 71T^{2} \) |
| 73 | \( 1 - 4.69T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 4.22T + 97T^{2} \) |
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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
−7.953442835216197703012287996102, −6.94248015126474000601611063321, −6.51129104859438612063132624386, −5.81697240435040144093399322635, −5.04929215177345380996092963765, −4.47911966338601397646067612997, −3.89968432594355222083957593664, −3.16426314045494961404268404176, −1.98231560551170896626062123489, −0.984186086388579302491341132107,
0.984186086388579302491341132107, 1.98231560551170896626062123489, 3.16426314045494961404268404176, 3.89968432594355222083957593664, 4.47911966338601397646067612997, 5.04929215177345380996092963765, 5.81697240435040144093399322635, 6.51129104859438612063132624386, 6.94248015126474000601611063321, 7.953442835216197703012287996102
