L(s) = 1 | − 2-s − 2·3-s − 4-s − 2·5-s + 2·6-s − 3·7-s + 3·8-s + 9-s + 2·10-s − 11-s + 2·12-s − 5·13-s + 3·14-s + 4·15-s − 16-s − 6·17-s − 18-s − 8·19-s + 2·20-s + 6·21-s + 22-s − 6·23-s − 6·24-s − 25-s + 5·26-s + 4·27-s + 3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.13·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.577·12-s − 1.38·13-s + 0.801·14-s + 1.03·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.83·19-s + 0.447·20-s + 1.30·21-s + 0.213·22-s − 1.25·23-s − 1.22·24-s − 1/5·25-s + 0.980·26-s + 0.769·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25751 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 2341 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.27345958720139, −15.80500951569142, −15.20356733659090, −14.70735679103314, −13.95204990042930, −13.16007963727968, −12.81215670353112, −12.44967202510566, −11.66344722363233, −11.29379866845009, −10.61024625810045, −10.20350550820871, −9.690225918248316, −9.028161852247612, −8.473127061525397, −7.919261517409008, −7.137818026996893, −6.796464044359082, −6.085222378745268, −5.408268920980588, −4.721290107802177, −4.168603075328746, −3.668028121016701, −2.497951991107593, −1.775878884561265, 0, 0, 0,
1.775878884561265, 2.497951991107593, 3.668028121016701, 4.168603075328746, 4.721290107802177, 5.408268920980588, 6.085222378745268, 6.796464044359082, 7.137818026996893, 7.919261517409008, 8.473127061525397, 9.028161852247612, 9.690225918248316, 10.20350550820871, 10.61024625810045, 11.29379866845009, 11.66344722363233, 12.44967202510566, 12.81215670353112, 13.16007963727968, 13.95204990042930, 14.70735679103314, 15.20356733659090, 15.80500951569142, 16.27345958720139