| L(s) = 1 | + 2-s + 4-s + 5-s + 2.08·7-s + 8-s + 10-s − 2.20·11-s − 4.28·13-s + 2.08·14-s + 16-s + 4.93·17-s + 4.93·19-s + 20-s − 2.20·22-s + 3.35·23-s + 25-s − 4.28·26-s + 2.08·28-s + 2.81·29-s − 2.28·31-s + 32-s + 4.93·34-s + 2.08·35-s − 0.727·37-s + 4.93·38-s + 40-s + 1.47·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.787·7-s + 0.353·8-s + 0.316·10-s − 0.664·11-s − 1.18·13-s + 0.556·14-s + 0.250·16-s + 1.19·17-s + 1.13·19-s + 0.223·20-s − 0.470·22-s + 0.699·23-s + 0.200·25-s − 0.840·26-s + 0.393·28-s + 0.521·29-s − 0.410·31-s + 0.176·32-s + 0.845·34-s + 0.352·35-s − 0.119·37-s + 0.800·38-s + 0.158·40-s + 0.230·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.943946023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.943946023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 2.08T + 7T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 - 4.93T + 19T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 - 2.81T + 29T^{2} \) |
| 31 | \( 1 + 2.28T + 31T^{2} \) |
| 37 | \( 1 + 0.727T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 + 9.38T + 47T^{2} \) |
| 53 | \( 1 - 6.16T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 - 4.08T + 61T^{2} \) |
| 71 | \( 1 - 4.84T + 71T^{2} \) |
| 73 | \( 1 + 0.544T + 73T^{2} \) |
| 79 | \( 1 - 4.83T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 - 2.20T + 97T^{2} \) |
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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
−7.72633626760727591116733084234, −7.51678074859552183376989279629, −6.62131211810840599409297430544, −5.66844285181768270962852543711, −5.10104356570960563610654813863, −4.78954313608282756113983000815, −3.56751063087084859015278106674, −2.83083279810865154989203454986, −2.03177459569741288596429218517, −0.973024210035254159172176342806,
0.973024210035254159172176342806, 2.03177459569741288596429218517, 2.83083279810865154989203454986, 3.56751063087084859015278106674, 4.78954313608282756113983000815, 5.10104356570960563610654813863, 5.66844285181768270962852543711, 6.62131211810840599409297430544, 7.51678074859552183376989279629, 7.72633626760727591116733084234
