L(s) = 1 | − 1.45·2-s − 2.35·3-s + 0.119·4-s + 5-s + 3.42·6-s − 7-s + 2.73·8-s + 2.54·9-s − 1.45·10-s − 0.281·12-s − 0.899·13-s + 1.45·14-s − 2.35·15-s − 4.22·16-s + 0.644·17-s − 3.71·18-s + 0.219·19-s + 0.119·20-s + 2.35·21-s + 2.75·23-s − 6.44·24-s + 25-s + 1.30·26-s + 1.06·27-s − 0.119·28-s + 0.429·29-s + 3.42·30-s + ⋯ |
L(s) = 1 | − 1.02·2-s − 1.36·3-s + 0.0598·4-s + 0.447·5-s + 1.40·6-s − 0.377·7-s + 0.967·8-s + 0.849·9-s − 0.460·10-s − 0.0813·12-s − 0.249·13-s + 0.389·14-s − 0.608·15-s − 1.05·16-s + 0.156·17-s − 0.874·18-s + 0.0504·19-s + 0.0267·20-s + 0.514·21-s + 0.574·23-s − 1.31·24-s + 0.200·25-s + 0.256·26-s + 0.204·27-s − 0.0226·28-s + 0.0797·29-s + 0.626·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 3 | \( 1 + 2.35T + 3T^{2} \) |
| 13 | \( 1 + 0.899T + 13T^{2} \) |
| 17 | \( 1 - 0.644T + 17T^{2} \) |
| 19 | \( 1 - 0.219T + 19T^{2} \) |
| 23 | \( 1 - 2.75T + 23T^{2} \) |
| 29 | \( 1 - 0.429T + 29T^{2} \) |
| 31 | \( 1 + 6.05T + 31T^{2} \) |
| 37 | \( 1 + 0.527T + 37T^{2} \) |
| 41 | \( 1 + 9.74T + 41T^{2} \) |
| 43 | \( 1 - 2.94T + 43T^{2} \) |
| 47 | \( 1 - 1.87T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 1.93T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 6.90T + 67T^{2} \) |
| 71 | \( 1 + 0.408T + 71T^{2} \) |
| 73 | \( 1 - 8.18T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 8.13T + 83T^{2} \) |
| 89 | \( 1 + 3.73T + 89T^{2} \) |
| 97 | \( 1 + 6.55T + 97T^{2} \) |
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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
−8.135647280784013470954603758594, −7.15597302012351391651811612667, −6.73817514751564128664133565609, −5.81739163308302821940225488121, −5.18936651974289352708392623363, −4.53038772581870466259053553423, −3.36869453388368774258915598453, −2.00909472192578321546940124314, −0.977007765476218454501834392000, 0,
0.977007765476218454501834392000, 2.00909472192578321546940124314, 3.36869453388368774258915598453, 4.53038772581870466259053553423, 5.18936651974289352708392623363, 5.81739163308302821940225488121, 6.73817514751564128664133565609, 7.15597302012351391651811612667, 8.135647280784013470954603758594