| L(s) = 1 | + 3-s − 4·7-s + 9-s − 4·11-s − 13-s + 4·17-s − 7·19-s − 4·21-s + 4·23-s + 27-s + 5·29-s − 4·31-s − 4·33-s − 9·37-s − 39-s − 5·41-s − 10·43-s + 3·47-s + 9·49-s + 4·51-s − 9·53-s − 7·57-s + 6·59-s + 4·61-s − 4·63-s − 7·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.970·17-s − 1.60·19-s − 0.872·21-s + 0.834·23-s + 0.192·27-s + 0.928·29-s − 0.718·31-s − 0.696·33-s − 1.47·37-s − 0.160·39-s − 0.780·41-s − 1.52·43-s + 0.437·47-s + 9/7·49-s + 0.560·51-s − 1.23·53-s − 0.927·57-s + 0.781·59-s + 0.512·61-s − 0.503·63-s − 0.855·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.082723953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082723953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−15.83702996379411, −15.59298901444096, −14.85305539108820, −14.47305602126669, −13.55917652491025, −13.26960679916003, −12.64329937314357, −12.43141565960364, −11.58095445718065, −10.54672474138258, −10.35273739415126, −9.854984586922741, −9.104165128621246, −8.574395657733950, −8.011452938240019, −7.228202414086625, −6.734735931882752, −6.140133890903392, −5.279210826590815, −4.741060638988591, −3.663583571312512, −3.239965473671610, −2.606746320426566, −1.785166696074903, −0.4114706710331682,
0.4114706710331682, 1.785166696074903, 2.606746320426566, 3.239965473671610, 3.663583571312512, 4.741060638988591, 5.279210826590815, 6.140133890903392, 6.734735931882752, 7.228202414086625, 8.011452938240019, 8.574395657733950, 9.104165128621246, 9.854984586922741, 10.35273739415126, 10.54672474138258, 11.58095445718065, 12.43141565960364, 12.64329937314357, 13.26960679916003, 13.55917652491025, 14.47305602126669, 14.85305539108820, 15.59298901444096, 15.83702996379411
