| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s − 6·11-s − 13-s − 2·14-s + 16-s − 19-s − 6·22-s + 6·23-s − 26-s − 2·28-s + 3·29-s + 8·31-s + 32-s + 37-s − 38-s + 9·41-s − 8·43-s − 6·44-s + 6·46-s − 3·47-s − 3·49-s − 52-s + 3·53-s − 2·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 1.80·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.229·19-s − 1.27·22-s + 1.25·23-s − 0.196·26-s − 0.377·28-s + 0.557·29-s + 1.43·31-s + 0.176·32-s + 0.164·37-s − 0.162·38-s + 1.40·41-s − 1.21·43-s − 0.904·44-s + 0.884·46-s − 0.437·47-s − 3/7·49-s − 0.138·52-s + 0.412·53-s − 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.318259231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.318259231\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.951216494535965536416791539001, −7.34617574466723208465457481665, −6.53783625597293748414760525078, −5.98755731320750691020583417045, −4.97292697009776353203159179509, −4.77339975748842402186204390328, −3.51025572119388762630800147242, −2.86766311773071163352228488676, −2.25274955722774686131502453309, −0.69229657903260327630254401205,
0.69229657903260327630254401205, 2.25274955722774686131502453309, 2.86766311773071163352228488676, 3.51025572119388762630800147242, 4.77339975748842402186204390328, 4.97292697009776353203159179509, 5.98755731320750691020583417045, 6.53783625597293748414760525078, 7.34617574466723208465457481665, 7.951216494535965536416791539001
