L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s − 2·9-s + 10-s − 12-s − 4·13-s + 14-s + 15-s + 16-s + 3·17-s + 2·18-s − 20-s + 21-s − 6·23-s + 24-s − 4·25-s + 4·26-s + 5·27-s − 28-s + 5·29-s − 30-s − 7·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s − 1.10·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.223·20-s + 0.218·21-s − 1.25·23-s + 0.204·24-s − 4/5·25-s + 0.784·26-s + 0.962·27-s − 0.188·28-s + 0.928·29-s − 0.182·30-s − 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−13.12934582056221, −12.53729405432918, −12.12737921871900, −11.87064007440830, −11.47725460694264, −10.88891308597074, −10.38981065892452, −10.00946188905986, −9.551739831589465, −9.113764097498212, −8.443415815067624, −8.007193411195673, −7.643643366856818, −7.142401163345853, −6.482644600521524, −6.163779165477561, −5.489799302423304, −5.212173791105506, −4.432993302213271, −3.849702278236453, −3.253884097927006, −2.656503222726002, −2.133170773110738, −1.354939203844880, −0.4909288584834376, 0,
0.4909288584834376, 1.354939203844880, 2.133170773110738, 2.656503222726002, 3.253884097927006, 3.849702278236453, 4.432993302213271, 5.212173791105506, 5.489799302423304, 6.163779165477561, 6.482644600521524, 7.142401163345853, 7.643643366856818, 8.007193411195673, 8.443415815067624, 9.113764097498212, 9.551739831589465, 10.00946188905986, 10.38981065892452, 10.88891308597074, 11.47725460694264, 11.87064007440830, 12.12737921871900, 12.53729405432918, 13.12934582056221