| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 4·11-s + 6·13-s + 14-s + 16-s + 4·17-s + 4·22-s + 4·23-s + 6·26-s + 28-s − 3·29-s + 7·31-s + 32-s + 4·34-s − 37-s + 7·41-s − 10·43-s + 4·44-s + 4·46-s − 13·47-s + 49-s + 6·52-s + 6·53-s + 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.852·22-s + 0.834·23-s + 1.17·26-s + 0.188·28-s − 0.557·29-s + 1.25·31-s + 0.176·32-s + 0.685·34-s − 0.164·37-s + 1.09·41-s − 1.52·43-s + 0.603·44-s + 0.589·46-s − 1.89·47-s + 1/7·49-s + 0.832·52-s + 0.824·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.818615920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.818615920\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.68757692852554563903225305844, −6.69554089760191313845387746215, −6.41365486847878872102506918669, −5.61097153140857345217780313709, −4.96100228411401603798874221601, −4.07093629678414088485831711532, −3.59788578802565999793693719028, −2.85021943672435871550208170622, −1.57808767885349547082784924824, −1.09313779313879282574238443916,
1.09313779313879282574238443916, 1.57808767885349547082784924824, 2.85021943672435871550208170622, 3.59788578802565999793693719028, 4.07093629678414088485831711532, 4.96100228411401603798874221601, 5.61097153140857345217780313709, 6.41365486847878872102506918669, 6.69554089760191313845387746215, 7.68757692852554563903225305844
