L(s) = 1 | + 2-s + 2.23·3-s + 4-s − 1.61·5-s + 2.23·6-s − 3.85·7-s + 8-s + 2.00·9-s − 1.61·10-s + 2.23·12-s − 0.618·13-s − 3.85·14-s − 3.61·15-s + 16-s − 1.76·17-s + 2.00·18-s + 19-s − 1.61·20-s − 8.61·21-s + 4.23·23-s + 2.23·24-s − 2.38·25-s − 0.618·26-s − 2.23·27-s − 3.85·28-s − 1.52·29-s − 3.61·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.29·3-s + 0.5·4-s − 0.723·5-s + 0.912·6-s − 1.45·7-s + 0.353·8-s + 0.666·9-s − 0.511·10-s + 0.645·12-s − 0.171·13-s − 1.03·14-s − 0.934·15-s + 0.250·16-s − 0.427·17-s + 0.471·18-s + 0.229·19-s − 0.361·20-s − 1.88·21-s + 0.883·23-s + 0.456·24-s − 0.476·25-s − 0.121·26-s − 0.430·27-s − 0.728·28-s − 0.283·29-s − 0.660·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 13 | \( 1 + 0.618T + 13T^{2} \) |
| 17 | \( 1 + 1.76T + 17T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 + 1.14T + 37T^{2} \) |
| 41 | \( 1 + 7.09T + 41T^{2} \) |
| 43 | \( 1 + 9.70T + 43T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 1.90T + 59T^{2} \) |
| 61 | \( 1 + 6.61T + 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 + 8.23T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.938311287615067027256450119284, −7.19067486828980283723250443888, −6.64884586895535107843347327122, −5.80450665565625033131947837239, −4.74843159085696755802413995431, −3.93915775758761204684414141910, −3.16863260806119009757599960044, −2.97583869090126255660477465309, −1.77857402604288117919910419113, 0,
1.77857402604288117919910419113, 2.97583869090126255660477465309, 3.16863260806119009757599960044, 3.93915775758761204684414141910, 4.74843159085696755802413995431, 5.80450665565625033131947837239, 6.64884586895535107843347327122, 7.19067486828980283723250443888, 7.938311287615067027256450119284