| L(s) = 1 | + 4-s + 5-s − 9-s + 11-s + 16-s − 2·19-s + 20-s + 25-s + 29-s − 36-s − 2·41-s + 44-s − 45-s − 49-s + 55-s − 2·59-s + 2·61-s + 64-s − 2·71-s − 2·76-s + 2·79-s + 80-s + 81-s − 2·95-s − 99-s + 100-s − 2·101-s + ⋯ |
| L(s) = 1 | + 4-s + 5-s − 9-s + 11-s + 16-s − 2·19-s + 20-s + 25-s + 29-s − 36-s − 2·41-s + 44-s − 45-s − 49-s + 55-s − 2·59-s + 2·61-s + 64-s − 2·71-s − 2·76-s + 2·79-s + 80-s + 81-s − 2·95-s − 99-s + 100-s − 2·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.582504040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.582504040\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.657690913124652638922065226163, −8.716575559254035293329566617057, −8.215132876172242646199215207641, −6.81710529128446831042603282026, −6.44203901459296729587559275647, −5.79638441243362708068291058136, −4.72870889080362041249294729875, −3.40929935936144731211902270224, −2.45419090862471360391264568739, −1.60345078524691969571309454163,
1.60345078524691969571309454163, 2.45419090862471360391264568739, 3.40929935936144731211902270224, 4.72870889080362041249294729875, 5.79638441243362708068291058136, 6.44203901459296729587559275647, 6.81710529128446831042603282026, 8.215132876172242646199215207641, 8.716575559254035293329566617057, 9.657690913124652638922065226163
