| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s + 3.75·13-s + 14-s + 16-s + 1.75·17-s − 8.12·19-s − 20-s − 22-s − 7.14·23-s + 25-s + 3.75·26-s + 28-s − 9.51·29-s + 6.12·31-s + 32-s + 1.75·34-s − 35-s − 5.75·37-s − 8.12·38-s − 40-s + 4.12·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 1.04·13-s + 0.267·14-s + 0.250·16-s + 0.426·17-s − 1.86·19-s − 0.223·20-s − 0.213·22-s − 1.49·23-s + 0.200·25-s + 0.737·26-s + 0.188·28-s − 1.76·29-s + 1.10·31-s + 0.176·32-s + 0.301·34-s − 0.169·35-s − 0.946·37-s − 1.31·38-s − 0.158·40-s + 0.644·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 3.75T + 13T^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 19 | \( 1 + 8.12T + 19T^{2} \) |
| 23 | \( 1 + 7.14T + 23T^{2} \) |
| 29 | \( 1 + 9.51T + 29T^{2} \) |
| 31 | \( 1 - 6.12T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 + 8.12T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 0.610T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 4.98T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 8.12T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 8.90T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−7.76104153668139530038318015318, −6.66490940519003090322982407158, −6.22315025920556225544951061987, −5.46660221404255909144738983369, −4.65177129203662840285166443731, −3.94816402854036180245483126776, −3.44261196050074013296157654432, −2.28445690334232451786152707223, −1.55207668086363353708547768096, 0,
1.55207668086363353708547768096, 2.28445690334232451786152707223, 3.44261196050074013296157654432, 3.94816402854036180245483126776, 4.65177129203662840285166443731, 5.46660221404255909144738983369, 6.22315025920556225544951061987, 6.66490940519003090322982407158, 7.76104153668139530038318015318
