L(s) = 1 | + 20.7·2-s + 116.·3-s − 79.6·4-s − 29.0·5-s + 2.42e3·6-s − 1.09e4·7-s − 1.23e4·8-s − 6.11e3·9-s − 603.·10-s + 1.40e4·11-s − 9.28e3·12-s − 3.90e4·13-s − 2.26e5·14-s − 3.37e3·15-s − 2.14e5·16-s − 3.16e5·17-s − 1.27e5·18-s + 8.56e5·19-s + 2.31e3·20-s − 1.27e6·21-s + 2.92e5·22-s + 2.51e6·23-s − 1.43e6·24-s − 1.95e6·25-s − 8.12e5·26-s − 3.00e6·27-s + 8.69e5·28-s + ⋯ |
L(s) = 1 | + 0.918·2-s + 0.830·3-s − 0.155·4-s − 0.0207·5-s + 0.762·6-s − 1.71·7-s − 1.06·8-s − 0.310·9-s − 0.0190·10-s + 0.290·11-s − 0.129·12-s − 0.379·13-s − 1.57·14-s − 0.0172·15-s − 0.820·16-s − 0.919·17-s − 0.285·18-s + 1.50·19-s + 0.00323·20-s − 1.42·21-s + 0.266·22-s + 1.87·23-s − 0.881·24-s − 0.999·25-s − 0.348·26-s − 1.08·27-s + 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + 1.87e6T \) |
good | 2 | \( 1 - 20.7T + 512T^{2} \) |
| 3 | \( 1 - 116.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 29.0T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.09e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.40e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.90e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.16e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.56e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.51e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.16e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.02e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + 7.28e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.46e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.46e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.03e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.17e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.39e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.76e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.58e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.77e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.52e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.87e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.77e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.35e9T + 7.60e17T^{2} \) |
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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
−13.55111685642879480488643057593, −13.08647198152043912157961450771, −11.71804556166452681114776434256, −9.622506797699085921880460581541, −8.946456390729419901102896867786, −6.96693469374191736841612054891, −5.51138408430608438116007424916, −3.67176701087675617548585023748, −2.85414154134567379323323563665, 0,
2.85414154134567379323323563665, 3.67176701087675617548585023748, 5.51138408430608438116007424916, 6.96693469374191736841612054891, 8.946456390729419901102896867786, 9.622506797699085921880460581541, 11.71804556166452681114776434256, 13.08647198152043912157961450771, 13.55111685642879480488643057593