L(s) = 1 | − 2-s + 0.912·3-s + 4-s − 1.77·5-s − 0.912·6-s + 5.04·7-s − 8-s − 2.16·9-s + 1.77·10-s + 0.912·12-s − 5.42·13-s − 5.04·14-s − 1.62·15-s + 16-s + 1.82·17-s + 2.16·18-s + 19-s − 1.77·20-s + 4.60·21-s + 5.61·23-s − 0.912·24-s − 1.84·25-s + 5.42·26-s − 4.71·27-s + 5.04·28-s − 2.23·29-s + 1.62·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.527·3-s + 0.5·4-s − 0.794·5-s − 0.372·6-s + 1.90·7-s − 0.353·8-s − 0.722·9-s + 0.562·10-s + 0.263·12-s − 1.50·13-s − 1.34·14-s − 0.418·15-s + 0.250·16-s + 0.442·17-s + 0.510·18-s + 0.229·19-s − 0.397·20-s + 1.00·21-s + 1.17·23-s − 0.186·24-s − 0.368·25-s + 1.06·26-s − 0.907·27-s + 0.953·28-s − 0.415·29-s + 0.296·30-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 | \( 1 - 0.912T + 3T^{2} \) |
| 5 | \( 1 + 1.77T + 5T^{2} \) |
| 7 | \( 1 - 5.04T + 7T^{2} \) |
| 13 | \( 1 + 5.42T + 13T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 23 | \( 1 - 5.61T + 23T^{2} \) |
| 29 | \( 1 + 2.23T + 29T^{2} \) |
| 31 | \( 1 - 0.386T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 9.28T + 41T^{2} \) |
| 43 | \( 1 + 5.45T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 6.85T + 53T^{2} \) |
| 59 | \( 1 - 3.70T + 59T^{2} \) |
| 61 | \( 1 + 2.89T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 0.872T + 71T^{2} \) |
| 73 | \( 1 + 9.78T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 - 4.28T + 83T^{2} \) |
| 89 | \( 1 + 5.60T + 89T^{2} \) |
| 97 | \( 1 + 2.90T + 97T^{2} \) |
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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
−7.957535315069225140390674567713, −7.48439083816546941922440321486, −7.02890727388534151889640548266, −5.49733819614987656645320138029, −5.11692080751253659325388970084, −4.17045708742979386695062889147, −3.15571016576172783087435960073, −2.29628110381337562636578656902, −1.42617288043383203285193359865, 0,
1.42617288043383203285193359865, 2.29628110381337562636578656902, 3.15571016576172783087435960073, 4.17045708742979386695062889147, 5.11692080751253659325388970084, 5.49733819614987656645320138029, 7.02890727388534151889640548266, 7.48439083816546941922440321486, 7.957535315069225140390674567713