L(s) = 1 | + 0.618·2-s + 3-s − 1.61·4-s + 0.618·6-s + 1.61·7-s − 2.23·8-s + 9-s − 1.61·12-s + 2.61·13-s + 1.00·14-s + 1.85·16-s − 4.23·17-s + 0.618·18-s − 0.236·19-s + 1.61·21-s − 3.85·23-s − 2.23·24-s + 1.61·26-s + 27-s − 2.61·28-s − 4.23·29-s − 7.85·31-s + 5.61·32-s − 2.61·34-s − 1.61·36-s + 8.94·37-s − 0.145·38-s + ⋯ |
L(s) = 1 | + 0.437·2-s + 0.577·3-s − 0.809·4-s + 0.252·6-s + 0.611·7-s − 0.790·8-s + 0.333·9-s − 0.467·12-s + 0.726·13-s + 0.267·14-s + 0.463·16-s − 1.02·17-s + 0.145·18-s − 0.0541·19-s + 0.353·21-s − 0.803·23-s − 0.456·24-s + 0.317·26-s + 0.192·27-s − 0.494·28-s − 0.786·29-s − 1.41·31-s + 0.993·32-s − 0.448·34-s − 0.269·36-s + 1.47·37-s − 0.0236·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 13 | \( 1 - 2.61T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 + 0.236T + 19T^{2} \) |
| 23 | \( 1 + 3.85T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 + 7.85T + 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 8.70T + 59T^{2} \) |
| 61 | \( 1 + 0.763T + 61T^{2} \) |
| 67 | \( 1 - 0.763T + 67T^{2} \) |
| 71 | \( 1 + 1.76T + 71T^{2} \) |
| 73 | \( 1 + 3.38T + 73T^{2} \) |
| 79 | \( 1 + 9.70T + 79T^{2} \) |
| 83 | \( 1 - 5.85T + 83T^{2} \) |
| 89 | \( 1 - 1.32T + 89T^{2} \) |
| 97 | \( 1 + 8.18T + 97T^{2} \) |
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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
−7.57475287483566565364646436900, −6.62069821216053124729633136333, −5.89651334799432474846262677999, −5.25572557331910140267308353940, −4.35774527107344163844783463491, −4.03798749105075112713196791113, −3.21583733705274006258453032334, −2.26156047156882474840053708483, −1.36369885284836127983785139961, 0,
1.36369885284836127983785139961, 2.26156047156882474840053708483, 3.21583733705274006258453032334, 4.03798749105075112713196791113, 4.35774527107344163844783463491, 5.25572557331910140267308353940, 5.89651334799432474846262677999, 6.62069821216053124729633136333, 7.57475287483566565364646436900