| L(s) = 1 | + 2·7-s − 11-s − 13-s + 17-s + 3·19-s − 4·23-s + 7·31-s − 37-s − 4·41-s + 7·43-s − 10·47-s − 3·49-s − 14·53-s − 10·59-s + 5·61-s − 10·71-s + 16·73-s − 2·77-s + 17·83-s − 7·89-s − 2·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s − 0.277·13-s + 0.242·17-s + 0.688·19-s − 0.834·23-s + 1.25·31-s − 0.164·37-s − 0.624·41-s + 1.06·43-s − 1.45·47-s − 3/7·49-s − 1.92·53-s − 1.30·59-s + 0.640·61-s − 1.18·71-s + 1.87·73-s − 0.227·77-s + 1.86·83-s − 0.741·89-s − 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.643090696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.643090696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.29148979829966, −11.97763784981756, −11.62198734358396, −11.04949446519994, −10.72703576084075, −10.13586156000089, −9.783705579617959, −9.312766970705040, −8.855990464502704, −8.048484294087911, −7.960170177876960, −7.680115810962308, −6.885192158769506, −6.390379574861144, −6.077317518018189, −5.290646984152943, −4.950475073231841, −4.658288562291867, −3.888053348729002, −3.463968460914198, −2.770743264536287, −2.377887200016484, −1.545380795018734, −1.282381099580666, −0.3237368800211062,
0.3237368800211062, 1.282381099580666, 1.545380795018734, 2.377887200016484, 2.770743264536287, 3.463968460914198, 3.888053348729002, 4.658288562291867, 4.950475073231841, 5.290646984152943, 6.077317518018189, 6.390379574861144, 6.885192158769506, 7.680115810962308, 7.960170177876960, 8.048484294087911, 8.855990464502704, 9.312766970705040, 9.783705579617959, 10.13586156000089, 10.72703576084075, 11.04949446519994, 11.62198734358396, 11.97763784981756, 12.29148979829966
