| L(s) = 1 | + 1.26·2-s + 1.37·3-s − 0.391·4-s + 1.74·6-s + 2.74·7-s − 3.03·8-s − 1.09·9-s − 11-s − 0.540·12-s + 2.90·13-s + 3.48·14-s − 3.06·16-s + 5.83·17-s − 1.39·18-s + 19-s + 3.78·21-s − 1.26·22-s − 1.00·23-s − 4.18·24-s + 3.68·26-s − 5.65·27-s − 1.07·28-s + 8.92·29-s + 5.91·31-s + 2.18·32-s − 1.37·33-s + 7.40·34-s + ⋯ |
| L(s) = 1 | + 0.896·2-s + 0.796·3-s − 0.195·4-s + 0.714·6-s + 1.03·7-s − 1.07·8-s − 0.365·9-s − 0.301·11-s − 0.155·12-s + 0.804·13-s + 0.930·14-s − 0.765·16-s + 1.41·17-s − 0.327·18-s + 0.229·19-s + 0.826·21-s − 0.270·22-s − 0.209·23-s − 0.854·24-s + 0.721·26-s − 1.08·27-s − 0.203·28-s + 1.65·29-s + 1.06·31-s + 0.385·32-s − 0.240·33-s + 1.26·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.070384555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.070384555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.26T + 2T^{2} \) |
| 3 | \( 1 - 1.37T + 3T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 23 | \( 1 + 1.00T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 - 5.91T + 31T^{2} \) |
| 37 | \( 1 - 8.15T + 37T^{2} \) |
| 41 | \( 1 + 7.98T + 41T^{2} \) |
| 43 | \( 1 + 6.88T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 5.29T + 53T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 - 7.63T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 0.653T + 71T^{2} \) |
| 73 | \( 1 + 4.93T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 0.982T + 97T^{2} \) |
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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
−8.286752180164567774333073379962, −7.81674984689396709571059070409, −6.52683033987302832263056494536, −5.93050387904260080248513138977, −5.00570921242114636235456446360, −4.70120159333387110673395844075, −3.48188736751327877544299540347, −3.22285385391455025536198017522, −2.16707796581308732662165554384, −0.956141513830355368203687979353,
0.956141513830355368203687979353, 2.16707796581308732662165554384, 3.22285385391455025536198017522, 3.48188736751327877544299540347, 4.70120159333387110673395844075, 5.00570921242114636235456446360, 5.93050387904260080248513138977, 6.52683033987302832263056494536, 7.81674984689396709571059070409, 8.286752180164567774333073379962
