| L(s) = 1 | + 2.73·2-s + 5.49·4-s − 0.170·5-s − 3.28·7-s + 9.58·8-s − 0.467·10-s + 11-s + 5.64·13-s − 9.00·14-s + 15.2·16-s + 4.19·17-s − 4.57·19-s − 0.939·20-s + 2.73·22-s + 1.02·23-s − 4.97·25-s + 15.4·26-s − 18.0·28-s + 9.58·29-s − 2.38·31-s + 22.5·32-s + 11.4·34-s + 0.561·35-s + 8.06·37-s − 12.5·38-s − 1.63·40-s − 4.54·41-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 2.74·4-s − 0.0763·5-s − 1.24·7-s + 3.38·8-s − 0.147·10-s + 0.301·11-s + 1.56·13-s − 2.40·14-s + 3.80·16-s + 1.01·17-s − 1.04·19-s − 0.209·20-s + 0.583·22-s + 0.212·23-s − 0.994·25-s + 3.03·26-s − 3.41·28-s + 1.77·29-s − 0.428·31-s + 3.98·32-s + 1.96·34-s + 0.0949·35-s + 1.32·37-s − 2.03·38-s − 0.258·40-s − 0.710·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.501645038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.501645038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 5 | \( 1 + 0.170T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 13 | \( 1 - 5.64T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 + 4.57T + 19T^{2} \) |
| 23 | \( 1 - 1.02T + 23T^{2} \) |
| 29 | \( 1 - 9.58T + 29T^{2} \) |
| 31 | \( 1 + 2.38T + 31T^{2} \) |
| 37 | \( 1 - 8.06T + 37T^{2} \) |
| 41 | \( 1 + 4.54T + 41T^{2} \) |
| 43 | \( 1 + 2.66T + 43T^{2} \) |
| 47 | \( 1 + 3.94T + 47T^{2} \) |
| 53 | \( 1 - 2.27T + 53T^{2} \) |
| 59 | \( 1 - 2.77T + 59T^{2} \) |
| 67 | \( 1 - 9.61T + 67T^{2} \) |
| 71 | \( 1 + 1.57T + 71T^{2} \) |
| 73 | \( 1 + 1.46T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 - 6.99T + 89T^{2} \) |
| 97 | \( 1 + 0.0351T + 97T^{2} \) |
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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
−7.86180184539340337380710662242, −6.86246951572585493873122103804, −6.33452371294998043037341482471, −6.05487719485581287925129437249, −5.20079131252180142903380425403, −4.27756083368156165002649854584, −3.64739923658271384391639406246, −3.21685704394120512215129099608, −2.28961986757774668072463799927, −1.14793026049885560513702599890,
1.14793026049885560513702599890, 2.28961986757774668072463799927, 3.21685704394120512215129099608, 3.64739923658271384391639406246, 4.27756083368156165002649854584, 5.20079131252180142903380425403, 6.05487719485581287925129437249, 6.33452371294998043037341482471, 6.86246951572585493873122103804, 7.86180184539340337380710662242
